Topological methods in algebraic geometry pdf
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Characteristic Classes
The goal of this lecture notes is to introduce to Characteristic Classes. This is an important tool of the contemporary mathematics, indispensable to work in geometry and topology, and also useful in… SHOWING 1-10 OF 105 REFERENCES SORT BYRelevanceMost Influenced PapersRecency Compact complex surfaces. Historical Note.- References.- The Content of the Book.- Standard Notations.- I. Preliminaries.- Topology and Algebra.- 1. Notations and Basic Facts.- 2. Some Properties of Bilinear forms.- 3. Vector… Lectures on Algebraic Topology I Preliminaries on Categories, Abelian Groups, and Homotopy.- x1 Categories and Functors.- x2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- x3 Homotopy.- II Homology of Complexes.-… Topological methods in moduli theory
One of the main themes of this long article is the study of projective varieties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such… Strong Rigidity of Locally Symmetric Spaces.
*Frontmatter, pg. i*Contents, pg. v* 1. Introduction, pg. 1* 2. Algebraic Preliminaries, pg. 10* 3. The Geometry of chi : Preliminaries, pg. 20* 4. A Metric Definition of the Maximal Boundary, pg.… Fibred Kähler and quasi-projective groups.
We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group πι^) of a compact Kahler manifold onto the fundamental group Π^ of a… Surfaces and the second homology of a group
LetG be a group andK(G, 1) an Eilenberg—MacLane space, i.e. π1(K(G,1))≅G, πi(K(G,1))=0,i≠1. We give a purely algebraic proof that the second homology groupH2(G)=H2(G,ℤ)≅H2(K(G,1)) is isomorphic to…
Bull. Math. Sci. (2015) 5:287–449 DOI 10.1007/s13373-015-0070-1 Topological methods in moduli theory F. Catanese1 Received: 5 November 2014 / Revised: 9 June 2015 / Accepted: 10 June 2015 / Published online: 18 August 2015 © The Author(s) 2015. This article is published with open access at SpringerLink.com Abstract One of the main themes of this long article is the study of projective vari- eties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera–de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)’s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)’s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov. The present work took place in the realm of the DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”. Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar. BF. Catanese 1Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, NW II, Universitätsstr. 30, 95447 Bayreuth, Germany 123
288 F. Catanese algebraic surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S. Keywords Moduli spaces ·Projective varieties ·Classifying spaces ·Group cohomology ·Group homology ·Symmetry marked moduli spaces ·Group of automorphisms ·Bagnera–de Franchis varieties ·Absolute Galois group Contents 1 Introduction ............................................ 289 2 Prehistory and beyond ...................................... 292 3 Algebraic topology: non existence and existence of continuous maps .............. 297 4 Projective varieties which are K(π, 1).............................. 303 5 A trip around Bagnera–de Franchis varieties and group actions on Abelian varieties ...... 307 5.1 Bagnera–de Franchis varieties .............................. 307 5.2 Actions of a finite group on an Abelian variety ..................... 309 5.3 The general case where Gis Abelian .......................... 311 5.4 Bagnera–de Franchis varieties of small dimension ................... 313 6 Orbifold fundamental groups and rational K(π , 1)’s ....................... 316 6.1 Orbifold fundamental group of an action ........................ 316 6.2 Rational K(π, 1)’s: basic examples ........................... 318 6.3 The moduli space of curves ............................... 319 6.4 Teichmüller space .................................... 320 6.5 Singularities of Mg,I .................................. 322 6.6 Group cohomology and equivariant cohomology .................... 326 6.7 Group homology, Hopf’s theorem, Schur multipliers .................. 332 6.8 Calculating H2(G,Z)via combinatorial group theory ................. 337 6.9 Sheaves and cohomology on quotients, linearizations ................. 339 6.10 Hodge bundles of weight =1 .............................. 345 6.11 A surface in a Bagnera–De Franchis threefold ..................... 346 7 Regularity of classifying maps and fundamental groups of projective varieties ......... 351 7.1 Harmonic maps ..................................... 351 7.2 Kähler manifolds and some archetypal theorem .................... 354 7.3 Siu’s results on harmonic maps ............................. 357 7.4 Hodge theory and existence of maps to curves ..................... 360 7.5 Restrictions on fundamental groups of projective varieties ............... 363 7.6 Kähler versus projective, Kodaira’s problem and Voisin’s negative answer ...... 367 7.7 The Shafarevich conjecture ............................... 373 7.8 Strong and weak rigidity for projective K(π, 1)manifolds ............... 380 7.9 Can we work with locally symmetric varieties? ..................... 386 8 Inoue type varieties ........................................ 386 9 Moduli spaces of surfaces and higher dimensional varieties ................... 394 9.1 Kodaira–Spencer–Kuranishi theory ........................... 395 9.2 Kuranishi and Teichmüller ............................... 398 9.3 Varieties with singularities ................................ 400 10 Moduli spaces of surfaces of general type ............................ 401 10.1 Canonical models of surfaces of general type. ..................... 401 10.2 The Gieseker moduli space ............................... 402 10.3 Components of moduli spaces and deformation equivalence .............. 403 10.4 Automorphisms and canonical models ......................... 405 10.5 Kuranishi subspaces for automorphisms of a fixed type ................ 408 11 Moduli spaces of symmetry marked varieties .......................... 410 11.1 Moduli marked varieties ................................. 410 11.2 Moduli of curves with automorphisms ......................... 412 11.3 Numerical and homological invariants of group actions on curves ........... 415 123
Topological methods in moduli theory 289 11.4 The refined homology invariant in the ramified case .................. 417 11.5 Genus stabilization of components of moduli spaces of curves with G-symmetry ... 420 11.6 Classification results for certain concrete groups .................... 422 11.7 Sing(Mg)II: loci of curves with automorphisms in Mg................ 424 11.8 Stable curves and their automorphisms, Si ng(Mg)................... 424 11.9 Branch stabilization and relation with other approaches ................ 426 11.10 Miller’s description of the second homology of a group and developments ...... 427 12 Connected components of moduli spaces and the action of the absolute Galois group ..... 428 12.1 Galois conjugates of projective classifying spaces ................... 429 12.2 Connected components of Gieseker’s moduli space .................. 430 12.3 Arithmetic of moduli spaces and faithful actions of the absolute Galois group ..... 432 12.4 Change of fundamental group .............................. 433 13 Stabilization results for the homology of moduli spaces of curves and Abelian varieties .... 434 13.1 Epilogue ......................................... 435 References ............................................... 436 1 Introduction The interaction of algebraic geometry and topology has been such, in the last three centuries, that it is often difficult to say when does a result belong to one discipline or to the other, the archetypical example being the Bézout theorem, first conceived through a process of geometrical degeneration (algebraic hypersurfaces degenerating to union of hyperplanes), and later clarified through topology and through algebra. This ‘caveat’ is meant to warn the reader that a more appropriate title for the present survey article could be: ‘Some topological methods in moduli theory, and from the personal viewpoint, taste and understanding of the author’. In fact, many topics are treated, some classical and some very recent, but with a choice converging towards some well defined research interests. I considered for some time the tempting and appealing title ‘How can the angel of topology live happily with the devil of abstract algebra’, paraphrasing the motto by Hermann Weyl.1 The latter title would have matched with my personal philosophical point of view: while it is reasonable that researchers in mathematics develop with enthusiasm and dedication new promising mathematical tools and theories, it is important then that the accumulated knowledge and cultural wealth (the instance of topology in the twentieth century being a major one) be not lost afterwards. This wealth must indeed not only be invested and exploited, but also further developed by addressing problems in other fields, problems which often raise new and fascinating questions. In more down to earth words, the main body of the article is meant to be an invitation for algebraic geometers to use more classical topology. This invitation is not new, see for istance the work of Atiyah and Bott on the moduli spaces of vector bundles on curves [15]; but explains the structure of the article which is, in a sense, that of a protracted colloquium talk, and where we hope that also topologists, for which many of these notions are well known, will get new kicks coming from algebraic geometry, and especially moduli theory. 1In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain [373], p. 500. My motto is instead: ‘Any good mathematical theory requires several good theorems. Conversely, a really good theorem requires several good theories. ’ 123
290 F. Catanese In this article we mostly consider moduli theory as the fine part of classification theory of complex varieties: and we want to show how in some lucky cases topology helps also for the fine classification, allowing the study of the structure of moduli spaces: as we have done quite concretely in several papers [30–33,35–37]. We have already warned the reader about the inhomogeneity of the level assumed in the text: usually many sections start with very elementary arguments but, at a certain point, when we deal with current problems, the required knowledge may raise considerably. Let us try to summarize the logical thread of the article. Algebraic topology flourished from some of its applications (such as Brouwer’s fixed point theorem, or the theorem of Borsuk–Ulam) inferring the non existence of certain continuous maps from the observation that their existence would imply the existence of homomorphisms satisfying algebraic properties which are manifestly impossible to be verified. Conversely, the theory of fibre bundles and homotopy theory give a topological incarnation of a group Gthrough its classifying space BG. The theory of classifying spaces translates then group homomorphisms into continuous maps to classifying spaces. For instance, in algebraic geometry, the theory of Albanese varieties can be understood as dealing with the case where G is free abelian and the classifying maps are holomorphic. For more general G, an important question is the one of the regularity of these classifying maps, such as harmonicity, addressed by Eells and Sampson, and their complex analyticity addressed by Siu and others. These questions, which were at the forefront of mathematical research in the last 40years, have powerful applications to moduli theory. After a general introduction directed towards a broader public, starting with clas- sical theorems by Zeuthen-Segre and Lefschetz, proceeding to classifying spaces and their properties, I shall concentrate on some classes of projective varieties which are classifying spaces for some group, providing several explicit examples. I discuss then locally symmetric varieties, and at a certain length the quotients of Abelian varieties by a cyclic group acting freely, which are here called Bagnera–De Franchis varieties. At this point the article becomes instructional, and oriented towards graduate stu- dents, and several important topics, like orbifold fundamental groups, Teichmüller spaces, moduli spaces of curves, group cohomology and homology are treated in detail (and a new proof of a classical theorem of Hopf is sketched). Then some applications are given to concrete problems in moduli theory, in partic- ular a new construction of surfaces with pg=q=1 is given. The next section is devoted to a preparation for the rigidity and quasi-rigidity properties of projective varieties which are classifying spaces (meaning that their moduli spaces are completely determined by their topology); in the section are recalled the by now classical results of Eells and Sampson, and Siu’s results about complex analyticity of harmonic maps, with particular emphasis on bounded domains and locally symmetric varieties. Other more elementary results, based on Hodge theory, the theorem of Castelnuovo- De Franchis, and on the explicit constructions of classifying spaces are explained in detail because of their importance for Kähler manifolds. We then briefly dis- 123
Topological methods in moduli theory 291 cuss Kodaira’s problem and Voisin’s counterexamples, then we dwell on fundamental groups of projective varieties, and on the Shafarevich conjecture. Afterwards we deal with several concrete investigations of moduli spaces, which in fact lead to some group theoretical questions, and to the investigation of moduli spaces of varieties with symmetries. Some key examples are: varieties isogenous to a product, and the Inoue-type vari- eties introduced in recent work with Ingrid Bauer: for these the moduli space is determined by the topological type. I shall present new results and open questions concerning this class of varieties. In the final part, after recalling basic results on complex moduli theory, we shall also illustrate the concept of symmetry marked varieties and their moduli, discussing the several reasons why it is interesting to consider moduli spaces of triples (X,G,α) where Xis a projective variety, Gis a finite group, and αis an effective action of Gon X.IfXis the canonical model of a variety of general type, then Gis acting linearly on some pluricanonical model, and we have a moduli space which is a finite covering of a closed subspace MGof the moduli space. In the case of curves we show how this investigation is related to the description of the singular locus of the moduli space Mg(for instance of its irreducible components, see [129]), and of its compactification Mg(see [113]). In the case of surfaces there is another occurrence of Murphy’s law, as shown in my joint work with Ingrid Bauer [33]: the deformation equivalence for minimal models S and for canonical models differs drastically (nodal Burniat surfaces being the easiest example). This shows how appropriate it is to work with Gieseker’s moduli space of canonical models of surfaces. In the case of curves, there are interesting relations with topology. Moduli spaces Mg(G)of curves with a group Gof automorphisms of a fixed topological type have a description by Teichmüller theory, which naturally leads to conjecture genus stabiliza- tion for rational homology groups. I will then describe two equivalent descriptions of the irreducible components of Mg(G), surveying known irreducibility results for some special groups. A new fine homological invariant was introduced in our joint work with Lönne and Perroni: it allows to prove genus stabilization in the ramified case, extend- ing a beautiful theorem due to Livingston [272] and Dunfield and Thurston [141], who dealt with the easier unramified case. Another important application is the following one, in the direction of arithmetic: in the 60’s Serre [336] showed that there exists a field automorphism σin the absolute Galois group Gal(¯ Q/Q), and a variety Xdefined over a number field, such that X and the Galois conjugate variety Xσhave non isomorphic fundamental groups, in particular they are not homeomorphic. In a joint paper with I. Bauer and F. Grunewald we proved a strong sharpening of this phenomenon discovered by Serre, namely, that if σis not in the conjugacy class of the complex conjugation then there exists a surface (isogenous to a product) Xsuch that Xand the Galois conjugate variety Xσhave non isomorphic fundamental groups. In the end we finish with an extremely quick mention of several interesting topics which we do not have the time to describe properly, among these, the stabilization results for the cohomology of moduli spaces and of arithmetic varieties. 123
292 F. Catanese 2 Prehistory and beyond The following discovery belongs to the 19-th century: consider the complex projective plane P2and two general homogeneous polynomials F,G∈C[x0,x1,x2]of the same degree d. Then F,Gdetermine a linear pencil of curves Cλ,∀λ=(λ0,λ 1)∈P1, Cλ:= x=(x0,x1,x2)∈P2|λ0F(x)+λ1G(x)=0. One sees that the curve Cλis singular for exactly μ=3(d−1)2values of λ, as can be verified by an elementary argument which we now sketch. In fact, xis a singular point of some Cλiff the following system of three homoge- neous linear equations in λ=(λ0,λ 1)has a nontrivial solution: λ0∂F ∂xi (x)+λ1∂G ∂xi (x)=0,∀i=0,1,2. By generality of F,Gwe may assume that the curves C0:= {x|F(x)=0}and C1:= {x|G(x)=0}are smooth and intersect transversally (i.e., with distinct tangents) in d2distinct points; hence if a curve of the pencil Cλhas a singular point x, then we may assume that for this point we have F(x)= 0= G(x), and then λis uniquely determined. If now ∂F ∂x0and ∂G ∂x0do not vanish simultaneously in x, then the above system has a nontrivial solution if and only if ∂G ∂x0·∂F ∂xi−∂F ∂x0·∂G ∂xi(x)=0,i=1,2. By the theorem of Bézout (see [369]) the above two equations have (2(d−1))2= 4(d−1)2solutions, including among these the (d−1)2solutions of the system of two equations ∂F ∂x0(x)=∂G ∂x0(x)=0. One sees that, for F,Ggeneral, there are no common solutions of the system ∂F ∂x0(x)=∂G ∂x0(x)=∂G ∂x1·∂F ∂x2−∂F ∂x1·∂G ∂x2(x)=0, hence the solutions of the above system are indeed 3(d−1)2=4(d−1)2−(d−1)2. It was found indeed that, rewriting μ=3(d−1)2=d2+2d(d−3)+3, the above formula generalizes to a beautiful formula, valid for any smooth algebraic surface S, and which is the content of the so-called theorem of Zeuthen-Segre; this goes as follows: observe in fact that d2is the number of points where the curves of the pencil meet, while the genus gof a plane curve of degree dequals (d−1)(d−2) 2. 123
Topological methods in moduli theory 293 Theorem 1 (Zeuthen-Segre, classical) Let S be a smooth projective surface, and let Cλ,λ∈P1be a linear pencil of curves of genus g which meet transversally in δdistinct points. If μis the number of singular curves in the pencil (counted with multiplicity), then μ−δ−2(2g−2)=I+4, where the integer I is an invariant of the algebraic surface, called Zeuthen-Segre invariant. Here, the integer δequals the self-intersection number C2of the curve C, while in modern terms the number 2g−2=C2+KS·C,KSbeing the divisor (zeros minus poles) of a rational differential 2-form. In particular, our previous calculation shows that for P2the invariant I=−1. The interesting part of the discovery is that the integer I+4 is not only an algebraic invariant, but is indeed a topological invariant. Indeed, for a compact topological space Xwhich can be written as the disjoint union of locally closed sets Xi,i=1,...r,homeomorphic to an Euclidean space Rni, one can define e(X):= r i (−1)ni, and indeed this definition is compatible with the more abstract definition e(X)= dim(X) j=0 (−1)jrank Hj(X,Z). For example, the plane P2=P2 Cis obtained from a point attaching C=R2and then C2=R4, hence e(P2)=3 and we verify that e(P2)=I+4. While for an algebraic curve Cof genus gits topological Euler–Poincaré charac- teristic, for short Euler number, equals e(C)=2−2g, since Cis obtained as the disjoint union of one point, 2garcs, and a 2-disk (think of the topological realization as the quotient of a polygon with 4gsides). The Euler Poincaré characteristic is multiplicative for products: e(X×Y)=e(X)·e(Y), and more generally for fibre bundles (a concept we shall introduce in the next section), and accordingly there is a generalization of the theorem of Zeuthen-Segre: Theorem 2 (Zeuthen-Segre, modern) Let S be a smooth compact complex surface, and let f :S→B be a fibration onto a projective curve B of genus b (i.e., the fibres f−1(P),P∈B, are connected), and denote by g the genus of the smooth fibres of f . Then 123
294 F. Catanese e(S)=(2b−2)(2g−2)+μ, where μ≥0, and μ=0if and only if all the fibres of f are either smooth or, in the case where g =1, a multiple of a smooth curve of genus 1. The technique of studying linear pencils turned out to be an invaluable tool for the study of the topology of projective varieties. In fact, Solomon Lefschetz in the beginning of the 20-th century was able to describe the relation holding between a smooth projective variety X⊂PNof dimension nand its hyperplane section W= X∩H, where His a general linear subspace of codimension 1, a hyperplane. The work of Lefschetz deeply impressed the Italian algebraic geometer Guido Castelnuovo, who came to the conclusion that algebraic geometry could no longer be carried over without the new emerging techniques, and convinced Oscar Zariski to go on setting the building of algebraic geometry on a more solid basis. The report of Zariski [379] had a big influence and the results of Lefschetz were reproven and vastly extended by several authors: they say essentially that homology and homotopy groups of real dimension smaller than the complex dimension nof Xare the same for Xands its hyperplane section W. In my opinion the nicest proofs of the theorems of Lefschetz are those given much later by Andreotti and Frankel [4,5]. Theorem 3 Let X be a smooth projective variety of complex dimension n, let W = X∩H be a smooth hyperplane section of X , and let further Y =W∩Hbe a smooth hyperplane section of W . First Lefschetz’ theorem: the natural homomorphism Hi(W,Z)→Hi(X,Z) is bijective for i <n−1, and surjective for i =n−1; the same is true for the natural homomorphisms of homotopy groups πi(W)→πi(X)(the results hold more generally, see [288], p. 41, even if X is singular and W contains the singular locus of X). Second Lefschetz’ theorem: The kernel of Hn−1(W,Z)→Hn−1(X,Z)is the subgroup Van Hn−1(W,Z)generated by the vanishing cycles, i.e., those cycles which are mapped to 0when W tends to a singular hyperplane section Wλin a pencil of hyperplane sections of X. Generalized Zeuthen-Segre theorem: if μis the number of singular hyperplane sections in a general linear pencil of hyperplane sections of X , then e(X)=2e(W)−e(Y)+(−1)nμ. Third Lefschetz’ theorem or Hard Lefschetz’ theorem: The first theorem and the universal coefficients theorem imply for the cohomology groups that H i(X,Z)→Hi(W,Z)is bijective for i <n−1, while H n−1(X,Z)→ Hn−1(W,Z)is injective. Defining Inv Hn−1(W,Z)as the Poincaré dual of the image of Hn−1(X,Z), then we have a direct sum decomposition (orthogonal for the cup product) after tensoring with Q: Hn−1(W,Q)=Inv Hn−1(W,Q)⊕Va n H n−1(W,Q). 123
Topological methods in moduli theory 295 Equivalently, the operator L given by cup product with the cohomology class h ∈ H2(X,Z)of a hyperplane, L :Hi(X,Z)→Hi+2(X,Z), induces an isomorphism Lj:Hn−j(X,Q)→Hn+j(X,Q), ∀j≤n. Not only the theorems of Lefschetz play an important role for our particular pur- poses, but we feel that we should also spend a few words sketching how they lead to some very interesting and still widely open conjectures, the Hartshorne conjectures (see [210]). Assume now that the smooth projective variety X⊂PNis the complete intersection of N−nhypersurfaces (this means the the sheaf IXof ideals of functions vanishing on Xis generated by polynomials F1,...,Fc,c:= N−nbeing the codimension of X). Then the theorems of Lefschetz2imply that the homology groups of Xequal those of PNfor i≤n−1 (recall that Hi(PN,Z)=0foriodd, while Hi(PN,Z)=Z for i≤2N,ieven). Similarly holds true for the homotopy groups, and we recall that, since PN=S2N+1/S1, then πi(PN)=0fori≤2N,i= 0,2. It was an interesting discovery by Barth (see [22–24,174] for the following and related results) that a similar (but weaker) result holds true for each smooth subvariety of PN, provided the codimension c=N−nof Xis smaller than the dimension. Theorem 4 (Barth–Larsen) Let X be a smooth subvariety of dimension n in PN: then the homomorphisms Hi(X,Z)→Hi(PN,Z), πi(X)→πi(PN) are bijective for i ≤n−c⇔i<2n−N+1, and surjective for i =n−c+1= 2n−N+1. Observe that, if N=n+1, then the above result yields exactly the one of Lefschetz, hence the theorem is sharp in this trivial case (but much weaker for complete intersec- tions of higher codimension). The case of the Segre embedding X:= P1×P2→P5is a case which shows how the theorem is sharp since (Z)2∼ =H2(X,Z)→H2(P8,Z)∼ = Zis surjective but not bijective. The reader might wonder why the theorem of Barth and Larsen is a generalization of the theorem of Lefschetz. First of all, while Barth used originally methods of holomorphic convexity in complex analysis (somehow reminiscent of Morse theory in the real case) Hartshorne showed [210] how the third Lefschetz Theorem implies the result of Barth for cohomology with coefficients in Q. Moreover a strong similarity with the Lefschetz situation follows from the fact that one may view it (as shown by Badescu, see [18]) as an application of the classical Lefschetz theorem to the intersection (X×PN)∩∼ =X, 2One uses here the following trick: the intersection of a projective variety Xwith a hypersurface is equal to the intersection of Xwith a hyperplane, but for a different embedding of X; this trick is used several times, starting with X=PN. 123
296 F. Catanese where ⊂PN×PNis the diagonal. In turn, an idea of Deligne [133,175]shows that the diagonal ⊂PN×PNbehaves ‘like’ a complete intersection, essentially because, under the standard birational map PN×PN P2N, it maps to a linear subspace of P2N. The philosophy is then that smooth subvarieties of small codimension behave like complete intersections. This could be no accident if the well known Hartshorne con- jecture [210] were true. Conjecture 5 (On subvarieties of small codimension, Hartshorne) Let X be a smooth subvariety of dimension n in PN, and assume that the dimension is bigger than twice the codimension, n >2(N−n): then X is a complete intersection. While the conjecture says nothing in the case of curves and surfaces and is trivial in the case where n≤4, since a codimension 1 subvariety is defined by a single equation, it starts to have meaning for n≥5 and c=N−n≥2. In the case where c=N−n=2, then by a result of Serre one knows (see [154], p. 143, also for a general survey of the Hartshorne conjecture for codimension 2 subvarieties) that Xis the zero set of a section sof a rank 2 holomorphic vector bundle Von PN(observe that in this codimension one has that Pic(X)∼ =Pic(PN), so that Xsatisfies the condition of being subcanonical: this means that ωX=OX(d) for some d): in the case c=2 the conjecture by Hartshorne is then equivalent to the conjecture Conjecture 6 (On vector bundles of rank 2 on projective space, Hartshorne) Let V be a rank 2 vector bundle on PN, and assume that N ≥7: then V is a direct sum of line bundles. The major evidence for the conjecture on subvarieties of small codimension comes from the concept of positivity of vector bundles ([175], also [267]). In fact, many construction methods of subvarieties Xof small codimension involve a realization of Xas the locus where a vector bundle map drops rank to an integer r, and Xbecomes singular if there are points of Xin the locus where the rank drops further down to (r−1). The expected dimension of is positive in the range of Hartshorne’s conjecture, but nevertheless this is not sufficient to show that ∩Xis non empty. Hartshorne’s Conjecture 5is related to projections: in fact, for each projective variety X⊂PNthere exists a linear projection PN P2n+1whose restriction to Xyields an embedding X⊂P2n+1: in other words, an embedding where the codimension is equal to the dimenion nplus 1. The condition of being embedded as a subvariety where the codimension is smaller or equal than the dimension is already a restriction (for instance, not all curves are plane curves, and smooth surfaces in P3 are simply connected by Lefschetz’s theorem), and the smaller the codimension gets, the stronger the restrictions are (as shown by Theorem 4). Speaking now in more technical terms, a necessary condition for Xto be a complete intersection is that the sheaf of ideals IXbe arithmetically Cohen-Macaulay (ACM, for short), which means that all higher cohomology groups Hi(PN,IX(d)) =0 vanish for n≥i>0 and ∀d∈Z. In view of the exact sequence 123
Topological methods in moduli theory 297 0→IX→OPN→OX→0 the CM condition amounts to two conditions: (1) Xis projectively normal, i.e., the linear system cut on Xby polynomials of degree dis complete ( H0(PN,OPN(d)) →H0(OX(d)) is surjective for all d≥0) (2) Hi(X,OX(d)) =0 for all d∈Z, and for all 0 <i<n=dim(X). In the case where Xhas codimension 2, and N≥6 (see [154], cor. 4.2, p. 165), the condition of being a complete intersection is equivalent to projective normality. Linear normality is the case d=1 and means that Xis not obtained as the projection of a non-degenerate variety from a higher dimensional projective space PN+1.This part of Hartshorne’s conjecture is the only one which has been verified: a smooth subvariety with n≥2c−1=2(N−n)−1 is linearly normal (theorem of Zak [378]). For higher d, one considers the so-called formal neighbourhood of X: denoting by N∨ X:= IX/I2 Xthe conormal bundle of X, one sees that, in order to show projective normality, in view of the exact sequence 0→Im+1 X(d)→Im X(d)→SymmN∨ X(d)∼ =Im X/Im+1 X(d)→0, a crucial role is played by the cohomology groups Hq(Symm(N∨ X)(d))). We refer the reader to [323] for a discussion of more general Nakano type vanishing statements of the form Hq(Symm(N∨ X)(d)⊗p X)(these could be implied by very strong curvature properties on the normal bundle, see also [267]). Observe finally that the theorems of Lefschetz have been also extended to the case of singular varieties (see [172,184]), but we shall not need to refer to these extensions in the present paper. 3 Algebraic topology: non existence and existence of continuous maps The first famous achievements of algebraic topology were based on functoriality, which was used to infer the nonexistence of certain continuous maps. The Brouwer’s fixed point theorem says that every continuous self map f:Dn→ Dn, where Dn={x∈Rn||x|≤1}is the unit disk, has a fixed point. The argument is by contradiction: otherwise, letting φ(x)be the intersection of the boundary Sn−1 of Dnwith the half line stemming from f(x)in the direction of x,φwouldbea continuous map φ:Dn→Sn−1,s.t.φ|Sn−1=IdSn−1. 123
298 F. Catanese The key point is to show that the reduced3homology group Hn−1(Sn−1,Z)∼ =Z, while Hn−1(Dn,Z)=0, the disc being contractible; after that, denoting by ι:Sn−1→Dn the inclusion, functoriality of homology groups, since φ◦ι=IdSn−1, would imply 0=Hn−1(φ) ◦Hn−1(ι) =Hn−1(IdSn−1)=IdZ, the desired contradiction. Also well known is the Borsuk–Ulam theorem, asserting that there is no odd con- tinuous function F:Sn→Smfor n>m(odd means that F(−x)=−F(x)). Here there are two ingredients, the main one being the cohomology algebra, and its contravariant functoriality: to any continuous map f:X→Ythere corresponds an algebra homomorphism f∗:H∗(Y,R)=⊕ dim(Y) i=0Hi(Y,R)→H∗(X,R), for any ring Rof coefficients. In our case one takes as X:= Pn R=Sn/{±1}, similarly Y:= Pm R=Sm/{±1} and lets fbe the continuus map induced by F. One needs to show that, choosing R=Z/2Z, then the cohomology algebra of real projective space is a truncated polynomial algebra, namely: H∗(Pn R,Z/2Z)∼ =(Z/2Z)[ξn]/(ξn+1 n). The other ingredient consists in showing that f∗([ξm])=[ξn], [ξm]denoting the residue class in the quotient algebra. One gets then the desired contradiction since, if n>m, 0=f∗(0)=f∗([ξm]m+1)=f∗([ξm])m+1=[ξn]m+1= 0. Notice that up to now we have mainly used that fis a continuous map f:= Pn R→Pm R, while precisely in order to obtain that f∗([ξm])=[ξn]we must make use of the hypothesis that fis induced by an odd function F. This property can be interpreted as the property that one has a commutative diagram Sn→Sm ↓↓ Pn R→Pm R which exhibits the two sheeted covering of Pn Rby Snas the pull-back of the analo- gous two sheeted cover for Pm R. Now, as we shall digress soon, any such two sheeted covering is given by a homomorphism of H1(X,Z/2Z)→Z/2Z, i.e., by an element 3The reduced homology group differs from the ordinary one only for i=0, and for i=0isdefinedas the kernel of the degree surjection onto Z: this distinction is only needed in order to treat the case n=1on an equal footing. 123
Topological methods in moduli theory 299 in H1(X,Z/2Z), and this element is trivial if and only if the covering is trivial (that is, homeomorphic to X×(Z/2Z), in other words a disconnected cover). This shows that the pull back of the cover, which is nontrivial, corresponds to f∗([ξm])and is nontrivial, hence f∗([ξm])=[ξn]. As we saw already in the first section, algebraic topology attaches to a good topolog- ical space homology groups Hi(X,R), which are covariantly functorial, a cohomology algebra H∗(X,R)which is contravariantly functorial, and these groups can be cal- culated, by virtue of the Mayer Vietoris exact sequence and of excision (see any textbook), by chopping the space in smaller pieces. In particular, these groups vanish when i>dim(X).ButtoXare also attached the homotopy groups πi(X). The com- mon feature is that homotopic maps induce the same homomorphisms on homology, cohomology, and homotopy. We are, for our purposes, more interested in the more mysterious homotopy groups, which, while not necessarily vanishing for i>dim(X), enjoy however a fundamental property. Recall the definition due to Whitney and Steenrod [350] of a fibre bundle. In the words of Steenrod, the notion of a fibre bundle is a weakening of the notion of a product, since a product X×Yhas two continuous projections pX:X×Y→X, and pY:X×Y→Y, while a fibre bundle Eover Bwith fibre Fhas only one projection, p=pB:E→Band its similarity to a product lies in the fact that for each point x∈Bthere is an open set Ucontaining x, and a homeomorphism p−1 B(U)∼ =U×F compatible with both projections onto U. The fundamental property of fibre bundles is that there is a long exact sequence of homotopy groups ···→πi(F)→πi(E)→πi(B)→πi−1(F)→πi−1(E)→πi−1(B)→··· where one should observe that πi(X)is a group for i≥1, an abelian group for i≥2, and for i=0 is just the set of arc-connected components of X(we assume the spaces to be good, that is, locally arcwise connected, semilocally simply connected, see [189], and, most of the times, connected). The special case where the fibre Fhas the discrete topology is the case of a covering space, which is called the universal covering if moreover π1(E)is trivial. Special mention deserves the following more special case. Definition 7 Assume that Eis arcwise connected, contractible (hence all homotopy groups πi(E)are trivial), and that the fibre Fis discrete, so that all the higher homotopy groups πi(B)=0fori≥2, while π1(B)∼ =π0(F)=F. Then one says that Bis a classifying space K(π , 1)for the group π=π1(B). In general, given a group π, a CW complex Bis said to be a K(π, 1)if πi(B)=0 for i≥2, while π1(B)∼ =π. Example 8 The easiest examples are the following ones, where (2) is a case where we have a complex projective variety (see Sect. 3for more such examples): (1) the real torus Tn:= Rn/Znis a classifying space K(Zn,1)for the group π=Zn; 123
300 F. Catanese (2) a complex projective curve Cof genus g≥2 is a classifying space K(πg,1), since by the uniformization theorem its universal covering is the Poincaré upper half plane H:= {z∈C|Im(z)>0}and its fundamental group π1(C)is isomorphic to the group πg:= α1,β 1,...α g,β g|g 1[αi,β i]=1, quotient of a free group with 2ggenerators by the normal subgroup generated by the relation g 1[αi,β i]; (3) a classifying space K(Z/2Z,1)is given by the inductivelimit P∞ R:= limn→∞ Pn R. To show this, it suffices to show that S∞:= limn→∞ Snis contractible or, equiv- alently, that the identity map is homotopic to a constant map. We do this as follows:4let σ:R∞→R∞be the shift operator, and first define a homotopy of the identity map of R∞\{0}to the constant map with value e1.The needed homotopy is the composition of two homotopies: F(t,v) := (1−t)v +tσ(v), 0≤t≤1, F(t,v) := (2−t)σ (v) +(t−1)e1,1≤t≤2,∀v∈R∞. Then we simply project the homotopy from R∞\{0}to S∞considering F(t,v) |F(t,v)|. These classifying spaces, although not unique, are unique up to homotopy- equivalence (we use the notation X∼h.e.Yto denote homotopy equivalence: this means that there exist continuous maps f:X→Y,g:Y→Xsuch that both compositions f◦gand g◦fare homotopic to the identity). Therefore, given two classifying spaces for the same group, they not only do have the same homotopy groups, but also the same homology and cohomology groups. Thus the following definition is well posed. Definition 9 Let be a finitely presented group, and let Bbe a classifying space for : then the homology and cohomology groups and algebra of are defined as Hi(, Z):= Hi(B, Z), Hi(, Z):= Hi(B, Z), H∗(, Z):= H∗(B, Z), and similarly for other rings of coefficients instead of Z. Remark 10 The concept of a classifying space BG is indeed more general: the group Gcould also be a Lie group, and then, if EG is a contractible space over which Ghas a free (continuous) action, then one defines BG := EG/G. The typical example is the simplest compact Lie group G=S1: then, keeping in mind that S1={z∈C||z|=1},wetakeasEG the space ES1:= S(C∞)=limn→∞S(Cn)=limn→∞ w∈Cn||w|=1. 4Following a suggestion of Marco Manetti. 123
Topological methods in moduli theory 301 Classifying spaces, even if often quite difficult to construct explicitly, are very important because they guarantee the existence of continuous maps! We have more precisely the following (cf. [349], Theorem 9, p. 427, and Theorem 11, p. 428) Theorem 11 Let Y be a ‘nice’ topological space, i.e., Y is homotopy-equivalent to a CW-complex, and let X be a nice space which is a K (π , 1)space: then, choosing base points y0∈Y,x0∈X , one has a bijective correspondence [(Y,y0), (X,x0)]∼ =Hom(π1(Y,y0), π1(X,x0)), [f] → π1(f), where [(Y,y0), (X,x0)]denotes the set of homotopy classes [f]of continuous maps f:Y→X such that f (y0)=x0(and where the homotopies F (y,t)are also required to satisfy F (y0,t)=x0,∀t∈[0,1]). In particular, the free homotopy classes [Y,X]of continuous maps are in bijective correspondence with the conjugacy classes of homomorphisms Hom(π1(Y,y0), π) (conjugation is here inner conjugation by Inn(π) on the target). Observe that, quite generally, the universal covering Eπof a classifying space Bπ:= K(π, 1)associates (by the lifting property) to a continuous map f:Y→Bπ aπ1(Y)-equivariant map ˜ f ˜ f:˜ Y→Eπ, where the action of π1(Y)on Eπis determined by the homomorphism ϕ:= π1(f): π1(Y)→π=π1(Bπ). Moreover, any ϕ:π1(Y)→π=π1(Bπ) determines a fibre bundle Eϕover Y with fibre Eπ: Eϕ:= (˜ Y×Eπ)/π1(Y)→Y=(˜ Y)/π1(Y)=Y, where the action of γ∈π1(Y)is as follows: γ(y,v) =(γ (y), ϕ (γ )(v)). While topology deals with continuous maps, when dealing with manifolds more regularity is wished for. For instance, when we choose for Ya differentiable manifold M, and the group πis abelian and torsion free, say π=Zr, then a more precise incarnation of the above theorem is given by the De Rham theory. In fact, a homomorphism ϕ:π1(Y)→Zrfactors through the Abelianization H1(Y,Z)of the fundamental group. Since H1(Y,Z)=Hom(H1(Y,Z), Z),ϕis equiv- alent to giving an element in ϕ∈H1(Y,Z)r⊂H1(Y,R)r∼ =H1 DR(Y,R)r, where H1 DR(Y,R)is the quotient space of the space of closed differentiable 1-forms modulo exact 1-forms. In this case the classifying space is a real torus Tr:= Rr/Zr. 123
302 F. Catanese Observe however that to give ϕ:π1(Y)→Zrit is equivalent to give its rcomponents ϕi,i=1,...,r, which are homomorphisms into Z, and giving a map to Tr:= Rr/Zr is equivalent to giving rmaps to T1:= R/Z: hence we may restrict ourselves to consider the case r=1. Let us sketch the basic idea of the previous Theorem 11 in this special case. Let us assume that Yis a cell complex, and define as usual Yjto be its j-th skeleton, the union of all the cells of dimension i≤j. Since the fundamental group of Yis generated by the free group F:= π1(Y1), we get a homomorphism :F→Zinducing ϕ. For each 1-cell γ∼ =S1we send γ→S1according to the map z∈S1→ zm∈S1, where m=(γ ). In this way we get a continuous map f1:Y1→S1, and we want to extend it inductively to Yjfor each j. Now, assume that fis already defined on Z, and that you are attaching an n-cell to Z, according to a continuous map ψ:∂(Dn)=Sn−1→Z. In order to extend fto Z∪ψDnit suffices to extend the map f◦ψto the interior of the disk Dn. This is possible once the map f◦ψ:Sn−1→S1is homotopic to a constant map. Now, for n=2, this condition holds by assumption: since ψ(S1)yields a relation for π1(Y), therefore its image under φmust be equal to zero. For higher n,n≥3, it suffices to observe that a continuous map h:Sn−1→S1 extends to the interior always: since Sn−1is simply connected, and S1=R/Z,hlifts to a continuous map h:Sn−1→R, and we can extend hto Dnby setting h(x):= |x|hx |x|,∀x= 0,h(0):= 0. (his then the composition of hwith the projection R→R/Z=S1). In general, when both Yand the classifying space X(asT1here) are differentiable manifolds, then each continuous map f:Y→Xis homotopic to a differentiable map f. Take in fact X⊂RNand observe that the implicit function theorem implies that there is a tubular neighbourhood X⊂TX⊂RNdiffeomorphic to a tubular neighbourhood of Xembedded as the 0-section of the normal bundle NXof the embedding X⊂RN. Therefore, approximating the function f:Y→RNby a differentiable function f with values in TX, we can use the bundle projection NX→Xto project f to a differentiable function f:Y→X, and similarly we can project the natural homotopy between fand f,f(y)+tf(y)to obtain a homotopy between fand f. Once we have a differentiable map f:Y→T1=R/Z, we simply take the lift ˜ f:˜ Y→R, and the differential d˜ fdescends to a closed differential form ηon Y such that its integral over a closed loop γis just φ(γ). We obtain the Proposition 12 Let Y be a differentiable manifold, and let X be a differentiable man- ifold that is a K (π, 1)space: then, choosing base points y0∈Y,x0∈X , one has a bijective correspondence [(Y,y0), (X,x0)]di f f ∼ =Hom(π1(Y), π ), [f] → π1(f), 123
Topological methods in moduli theory 303 where [(Y,y0), (X,x0)]di f f denotes the set of differential homotopy classes [f]of differentiable maps f :Y→X such that f (y0)=x0. In the case where X is a torus T r=Rr/Zr, then f is obtained as the projection onto T rof ˜ φ(y):= y y0 (η1,...,η r), η j∈H1(Y,Z)⊂H1 DR(Y,R). Remark 13 In the previous proposition, ηjis indeed a closed 1-form, representing a certain De Rham cohomology class with integral periods ( i.e., γηj=ϕ(γ ) ∈ Z,∀γ∈π1(Y)). Therefore fis defined by y y0(η1,...,η r)mod(Zr). Moreover, changing ηjwith another form ηj+dFjin the same cohomology class, one finds a homotopic map, since y y0(η j+tdFj)=y y0(η j)+t(Fj(y)−Fj(y0)). Before we dwell into a review of results concerning higher regularity of the classi- fying maps, we consider in the next section the basic examples of projective varieties that are classifying spaces. 4 Projective varieties which are K(π, 1) The following are the easiest examples of projective varieties which are K(π , 1)’s. (1) Projective curves Cof genus g≥2. By the Uniformization theorem, these have the Poincaré upper half plane H:= {z∈C|Im(z)>0}as universal covering, hence they are compact quotients C= H/, where ⊂PSL(2,R)is a discrete subgroup isomorphic to the fundamental group of C,π1(C)∼ =πg.Here πg:= α1,β 1,...α g,β g|g 1[αi,β i]=1 contains no elements of finite order. Hence, given a faithful action of πgon H,itfollows that necessarily acts freely on H. Moreover, the quotientmust be compact, otherwise Cwould be homotopically equivalent to a bouquet of circles, hence H2(C,Z)=0, a contradiction, since H2(C,Z)∼ =H2(πg,Z)∼ =Z, as one sees taking the standard realization of a classifying space for πgby glueing the 2gsides of a polygon in the usual pattern. Moreover, the complex orientation of Cinduces a standard generator [C]of H2(C,Z)∼ =Z, the so-called fundamental class. (2) AV : = Abelian varieties. More generally, a complex torus X=Cg/, where is a discrete subgroup of maximal rank (isomorphic then to Z2g), is a Kähler classifying space K(Z2g,1),the Kähler metric being induced by the translation invariant Euclidean metric i 2g 1dzj⊗ dzj. For g=1 one gets in this way all projective curves of genus g=1; but, for g>1, Xis in general not projective: it is projective, and called then an Abelian variety, if it satisfies the Riemann bilinear relations. These amount to the existence of a positive 123
304 F. Catanese definite Hermitian form Hon Cgwhose imaginary part A( i.e., H=S+iA), takes integer values on ×. In modern terms, there exists a positive line bundle Lon X, with Chern class A∈H2(X,Z)=H2(, Z)=∧ 2(Hom(, Z)), whose curvature form, equal to H, is positive (the existence of a positive line bundle on a compact com- plex manifold Ximplies that Xis projective algebraic, by Kodaira’s theorem, [247]). We shall indeed see (82) that Abelian varieties are exactly the projective K(π, 1) varieties, for which πis an abelian group. (3) LSM : = Locally symmetric manifolds. These are the quotients of a bounded symmetric domain Dby a cocompact discrete subgroup ⊂Aut (D)acting freely. Recall that a bounded symmetric domain Dis a bounded domain D⊂⊂ Cnsuch that its group Aut(D)of biholomorphisms contains for each point p∈D, a holomorphic automorphism σpsuch that σp(p)=p, and such that the derivative of σpat pis equal to −Id. This property implies that σ is an involution (i.e., it has order 2), and that Aut(D)0(the connected component of the identity) is transitive on D, and one can write D=G/K, where Gis a connected Lie group, and Kis a maximal compact subgroup. The two important properties are: (3.1) Dsplits uniquely as the product of irreducible bounded symmetric domains. (3.2) each such Dis contractible, since there is a Lie subalgebra Lof the Lie algebra Gof Gsuch that the exponential map is a homeomorphism L∼ =D. Hence Xis a classifying space for the group ∼ =π1(X). Bounded symmetric domains were classified by Elie Cartan [77], and there is only a finite number of them (up to isomorphism) for each dimension n. Recall the notation for the irreducible domains: (i) In,pis the domain D={Z∈Mat(n,p,C):Ip−tZ·Z>0}. (ii) II nis the intersection of the domain In,nwith the subspace of skew symmetric matrices. (iii) III nis instead the intersection of the domain In,nwith the subspace of symmetric matrices. (iv) The Cartan–Harish Chandra realization of a domain of type IV nin Cnis the subset Ddefined by the inequalities (compare [213], p. 527) |z2 1+z2 2+···+z2 n|<1, 1+|z2 1+z2 2+···+z2 n|2−2|z1|2+|z2|2+···+|zn|2>0. (v) D16 is the exceptional domain of dimension d=16. (vi) D27 is the exceptional domain of dimension d=27. We refer the reader to [213], Theorem 7.1, p. 383 and exercise D, pp. 526–527, and [329] p. 525 for a list of these irreducible bounded symmetric domains, and a description of all of them as homogeneous spaces G/K. In this context the domains are also called Hermitian symmetric spaces of non compact type. Each of these is contained in the so-called compact dual, which is a Hermitian symmetric spaces of compact type. The easiest example is, for type I, the Grassmann manifold. For type IV, the compact dual of Dis the hyperquadric Qn⊂Pn+1defined by the polynomial n−1 j=0X2 j−X2 n− 123
Topological methods in moduli theory 305 X2 n+1. Notice that SO0(n,2)⊂Aut(Qn). The Borel embedding j:D→Qnis given by j(z1,...,zn)=[2z1:2z2:···:2zn:i( −1):+1], where := z2 1+···+z2 n.Themap jidentifies the domain Dwith the SO0(n,2)-orbit of the point [0:0:···:1:i]∈Qn, i.e. D∼ =SO0(n,2)/SO(n)×SO(2). Among the bounded symmetric domains are the so called bounded symmetric domains of tube type, those which are biholomorphic to a tube domain, a generalized Siegel upper half-space TC=V⊕√−1C where Vis a real vector space and C⊂Vis a symmetric cone, i.e., a self dual homogeneous convex cone containing no full lines. In the case of type III domains, the tube domain is Siegel’s upper half space: Hg:= τ∈Mat(g,g,C)|τ=tτ,Im(τ ) > 0, a generalisation of the upper half-plane of Poincaré. Borel proved in [57] that for each bounded symmetric domain Dthere exists a com- pact free quotient X=D/, called a compact Clifford–Klein form of the symmetric domain D. A classical result of Hano (see [207] Theorem IV, p. 886, and Lemma 6.2, p. 317 of [292]) asserts that a bounded homogeneous domain that is the universal cover of a compact complex manifold is symmetric. (4) A particular, but very explicit case of locally symmetric manifolds is given by the VIP : = Varieties isogenous to a product. These were studied in [96], and they are defined as quotients X=(C1×C2×···×Cn)/ G of the product of projective curves Cjof respective genera gj≥2 by the action of a finite group Gacting freely on the product. In this case the fundamental group of Xis not so mysterious and fits into an exact sequence 1→π1(C1×C2×···×Cn)∼ =πg1×···×πgn→π1(X)→G→1. Such varieties are said to be of the unmixed type if the group Gdoes not permute the factors, i.e., there are actions of Gon each curve such that γ(x1,...,xn)=(γ x1,...,γxn), ∀γ∈G. Equivalently, each individual subgroup πgjis normal in π1(X). 123
306 F. Catanese (5) Kodaira fibrations f:S→B. Here Sis a smooth projective surface and all the fibres of fare smooth curves of genus g≥2, in particular fis a differentiable fibre bundle. Unlike the examples above, where C=(C1×C2)/G→C2/Gis a holomorphic fibre bundle with fibre C1if Gacts freely on C2, the second defining property for Kodaira fibrations is that the fibres are not all biholomorphic to each other. These Kodaira fibred surfaces Sare very interesting topological objects: they were constructed by Kodaira [251] as a counterexample to the conjecture that the index (of the cup product in middle cohomology) would be multiplicative for fibre bundles. In fact, for curves the pairing H1(C,Z)×H1(C,Z)→H2(C,Z)∼ =Zis skew symmetric, hence it has index zero; while Kodaira showed that the index of the cup product H2(S,Z)×H2(S,Z)→H4(S,Z)∼ =Zis strictly positive. The fundamental group of Sfits obviously into an exact sequence: 1→πg→π1(S)→πb→1, where gis the fibre genus and bis the genus of the base curve B, and it is known that b≥2, g≥3(see[109], also for more constructions and a thorough discussion of their moduli spaces). By simultaneous uniformization ([40]) the universal covering ˜ Sof a Kodaira fibred surface Sis biholomorphic to a bounded domain in C2(fibred over the unit disk := {z∈C||z|<1}with fibres isomorphic to ), which is not homogeneous. (6) Hyperelliptic surfaces: these are the quotients of a complex torus of dimension 2 by a finite group Gacting freely, and in such away that the quotient is not again a complex torus. These surfaces were classified by Bagnera and de Franchis ([19],seealso[21,156]) and they are obtained as quotients (E1×E2)/Gwhere E1,E2are two elliptic curves, and Gis an abelian group acting on E1by translations, and on E2effectively and in such a way that E2/G∼ =P1. (7) In higher dimension we define the Generalized Hyperelliptic Varieties (GHV) as quotients A/Gof an Abelian Variety Aby a finite group Gacting freely, and with the property that Gis not a subgroup of the group of translations. Without loss of generality one can then assume that Gcontains no translations, since the subgroup GTof translations in Gwould be a normal subgroup, and if we denote G=G/GT, then A/G=A/G, where Ais the Abelian variety A:= A/GT. We propose instead the name Bagnera–de Franchis (BdF) Varieties for those quotients X=A/Gwere Gcontains no translations, and Gis a cyclic group of order m, with generator g(observe that, when Ahas dimension n=2, the two notions coincide, thanks to the classification result of Bagnera and De Franchis [19]). A concrete description of such Bagnera–De Franchis varieties shall be given in the following section. We end this section giving an example of a projective variety which is not a K(π, 1), thus showing that the property of being a projective classifying space is lost after taking hyperplane sections. Proposition 14 Let n ≥3, and consider X =C1×···×Cn, the product of n projective curves Ciof respective genera gi≥1. Let X ⊂PNbe a projective embedding, and 123
Topological methods in moduli theory 307 let S be a smooth surface, obtained taking the complete intersection of X with n −2 hypersurfaces. Then π2(S)= 0, in particular, S is not a projective classifying space. Proof By the theorem of Lefschetz π1(S)∼ =π1(X), hence the universal covering ˜ Sof Sis a closed complex submanifold of ˜ X=Cr×Hn−r. Hence ˜ Sis a Stein manifold, therefore (see [4]) it has the homotopy type of a CW complex of real dimension ≤2. Since π1(˜ S)=0, we claim that H2(˜ S,Z)=π2(˜ S)=π2(S)= 0(thefirst isomorphism follows from the theorem of Hurewicz). Otherwise, all homology and homotopy groups of ˜ Swould be trivial and ˜ Swould be contractible, SwouldbeaK(π, 1), hence H∗(S,Z)=H∗(π1(S), Z)=H∗(X,Z). This is a contradiction, since H2n(X,Z)= 0,H2n(S,Z)=0, by our hypothesis that n≥3. 5 A trip around Bagnera–de Franchis varieties and group actions on Abelian varieties 5.1 Bagnera–de Franchis varieties Let A/Gbe a Generalized Hyperelliptic Variety. An easy observation is that any g∈G is induced by an affine transformation x→ αx+bon the universal cover, hence it does not have a fixed point on A=V/ if and only if there is no solution of the equation ∃x∈V,g(x)≡x(mod ) ⇔∃x∈V,λ ∈, (α −Id)x=λ−b. This remark implies that 1 must be an eigenvalue of αfor all non trivial transformations g∈G. Since α∈GL() ∼ =GL(2n,Z), the eigenspace V1=Ker(α −Id)is a complex subspace defined over Q, hence we have an Abelian subvariety A1⊂A,A1:= V1/1, 1:= (V1∩). While what we said up to now was valid for any complex torus, we replace this assumption by the stronger assumption that Ais an Abelian variety. This assumption allows us to invoke Poincaré’ s complete reducibility theorem, so that we can split V=V1⊕V2,V2:= (V1)⊥,A2:= V2/2, 2:= (V2∩). We get then an isogeny A1×A2→A,with kernel T:= /(1⊕2). Observe that T∩Aj={0},j=1,2. In view of the splitting V=V1⊕V2, we can write, after a change of the origin in the affine space V2, α(v1+v2)=v1+α2v2,g(v) =v1+b1+α2v2,b1∈V1,b1/∈. 123
308 F. Catanese If α2has order equal to m, then necessarily the image of b1has order exactly m in A, by virtue of our assumption that Gcontains no translations. In other words, b1 induces a translation on A1of order exactly m. Now, glifts naturally to A1×A2,by g(a1,a2)=(a1+[b1],α 2a2), where [b1]is the class of b1in A1. We reach the conclusion that X=A/G=((A1×A2)/T)/G, where Tis the finite group of translations T=/(1⊕2). Conversely, given such an automorphism gof A1×A2, it descends to A:= (A1× A2)/Tif and only if the linear part of gsends Tto T. Denote now by Tjthe (isomorphic) image of T→Aj: then T⊂T1×T2is the graph of an isomorphism φ:T1→T2,hence the condition which allows gto descend to Ais that: (∗∗)(Id ×α2)(T)=T⇔α2◦φ=φ⇔ ⇔(α2−Id)◦φ=0⇔(α2−Id)T2=0. We are in the position to illustrate the standard example, before we give the more general more complete description of BdF varieties. The basic example is the one where m=2, hence α2is scalar multiplication by −1. Then φ=−φimplies that T2,T1are 2-torsion subgroups. Then also 2[b1]=0 implies that [b1]is a 2-torsion element. However [b1]cannot belong to T1,elseg:A→A would be induced by (a1,a2)→ (a1+[b1],α 2a2)≡(a1,α 2a2−φ([b1])) (mod T), which has a fixed point on A. We conclude that the standard example of Bagnera–de Franchis Varieties of order m=2 is the following: X=A/g,where A:= (A1×A2)/T,and T={(t1,φ(t1))},Tj⊂Aj[2],φ :T1∼ =T2,β 1∈A1[2]\T1, g:A→A,g(a1,a2)=(a1+β1,−a2). In order to conclude appropriately the above discussion, we give some useful defi- nition. Definition 15 We define first a Bagnera–de Franchis manifold (resp.: variety) of prod- uct type as a quotient X=A/Gwhere A=A1×A2,A1,A2are complex tori (resp.: Abelian Varieties), and G∼ =Z/mis a cyclic group operating freely on A, generated by an automorphism of the form g(a1,a2)=(a1+β1,α 2(a2)), 123
Topological methods in moduli theory 309 where β1∈A1[m]is an element of order exactly m, and similarly α2:A2→A2is a linear automorphism of order exactly mwithout 1 as eigenvalue (these conditions guarantee that the action is free). If moreover all eigenvalues of α2are primitive m-th roots of 1, we shall say that X=A/Gis a primary Bagnera–de Franchis manifold. We have the following proposition, giving a characterization of Bagnera- De Fran- chis varieties. Proposition 16 Every Bagnera–de Franchis variety X =A/G, where G ∼ =Z/m contains no translations, is the quotient of a Bagnera–de Franchis variety of product type, (A1×A2)/G by any finite subgroup T of A1×A2which satisfies the following properties: (1) T is the graph of an isomorphism between two respective subgroups T1⊂ A1,T2⊂A2, (2) (α2−Id)T2=0 (3) if g(a1,a2)=(a1+β1,α 2(a2)), then the subgroup of order m generated by β1 intersects T1only in {0}. In particular, we may write X as the quotient X =(A1×A2)/(G×T)by the abelian group G ×T. 5.2 Actions of a finite group on an Abelian variety Assume that we have the action of a finite group Gon a complex torus A=V/. Since every holomorphic map between complex tori lifts to a complex affine map of the respective universal covers, we can attach to the group Gthe group of affine transformations which fits into the exact sequence: 0→→→G→1. consists of all affine transformations of Vwhich lift transformations of the group G. Define now G0to be the subgroup of Gconsisting of all the translations in G. Proposition 17 The abstract group determines an exact sequence 0→→→G→1 such that ⊂,is a lattice in V , / =G0,G ⊂Aut (V/)contains no translations. Proof It is clear that V=⊗ZRas a real vector space, and we denote by VQ:= ⊗Q. Consider the homomorphism αLassociating to an affine transformation its linear part, and let := ker(αL:→GL(VQ)⊂GL(V)), G1:= Im(αL:→GL(VQ)). 123
310 F. Catanese is a subgroup of the group of translations in V, hence it is obviously Abelian, and maps isomorphically onto a lattice of Vwhich contains . We shall identify this lattice with , writing with shorthand notation ⊂V. In turn V=⊗ZR, and, if G:= /, then G∼ =G1and we have the exact sequence 0→→→G→1, yielding an embedding G⊂GL(). There remains to show that is determined by as an abstract group, indepen- dently of the exact sequence we started with. In fact, one property of is that it is a maximal abelian subgroup, normal and of finite index. Assume that has the same property: then their intersection 0:= ∩ is a normal subgroup of finite index, in particular 0⊗ZR=⊗ZR=V; hence ⊂ker(αL:→GL(V)) =, where αLis induced by conjugation on 0By maximality =. In the case where Ais an Abelian variety, we can say a little bit more about the affine representation of the group . Lemma 18 Let A an Abelian variety, and let G be a finite group of automorphisms. Then there is a positive integer r such that the group of affine transformations which are lifts of transformations in G satisfies ⊂Af f 1 r. In other words, we can write any transformation g ∈G as induced by the affine transformation of V : v→ α(v) +λ r,λ ∈. Proof Let Lbe an ample divisor class on A. Since Gis finite, there is a G-invariant such class L(simply replace Lby g∈Gg∗(L)=: L). We can further assume that the class of Lis not only G-invariant , but also indivisible. There is a homomorphism φL:A→A∨, such that φL(a):= T∗ aL−L∈Pic0(A)= A∨, and where Tais the translation by a,v→ v+a. Let KL:= ker(L), which is a finite group, and denote by rits exponent (hence, rKL=0 and KL⊂(1 r)/). Represent now Lby a line bundle whose cocycle is in Appell-Humbert normal form fλ(z)=ρ(λ)ex p(π H(z,λ) +π 2H(λ, λ)), and write a transformation g∈G as induced by v→ α(v) +b. The condition that the Chern class of Lis G-invariant implies that the Hermitian form Hon Vis left invariant by α(i.e., we have a group of isometries of Vfor the metric given by the Hermitian form H). Hence also the translation part leaves Linvariant, therefore the class of blies in KL⊂(1 r)/. Observe now that, given an affine group ⊂Af f ( ⊗ZR)as above, in order to obtain the structure of a complex torus on V/, we must give a complex structure on Vwhich makes the action of G∼ =G1complex linear. 123
Topological methods in moduli theory 311 In order to study the moduli spaces of the associated complex manifolds, we intro- duce therefore a further invariant, called Hodge type, according to the following definition. Definition 19 (i) Given a faithful representation G→Aut(), where is a free abelian group of even rank 2n,aG-Hodge decomposition is a G-invariant decomposition ⊗C=H1,0⊕H0,1,H0,1=H1,0. (ii) Write ⊗Cas the sum of isotypical components ⊗C=⊕ χ∈Irr(G)Uχ. Write also Uχ=Wχ⊗Mχ, where Wχis the irreducible representation corre- sponding to the character χ, and Mχis a trivial representation whose dimension is denoted nχ. Write accordingly V:= H1,0=⊕ χ∈Irr(G)Vχ,where Vχ=Wχ⊗M1,0 χ. Then the Hodge type of the decomposition is the datum of the dimensions ν(χ) := dimCM1,0 χ corresponding to the Hodge summands for non real representations (observe in fact that one must have: ν(χ) +ν( ¯χ) =dim(Mχ)). Remark 20 Given a faithful representation G→Aut (), where is a free abelian group of even rank 2n, all the G-Hodge decompositions of a fixed Hodge type are parametrized by an open set in a product of Grassmannians. Since, for a non real irreducible representation χone may simply choose M1,0 χto be a complex subspace of dimension ν(χ) of Mχ, and for Mχ=Mχ, one simply chooses a complex subspace M1,0 χof middle dimension. Then the open condition is just that (since M0,1 χ:= M1,0 χ) we want Mχ=M1,0 χ⊕M0,1 χ, or, equivalently, Mχ=M1,0 χ⊕M1,0 ¯χ. 5.3 The general case where Gis Abelian Assume now that Gis Abelian, and consider the linear representation ρ:G→GL(V). We get a splitting V=⊕ χ∈G∨Vχ=⊕ χ∈XVχ, 123
312 F. Catanese where G∨:= Hom(G,C∗)={χ:G→C∗}is the dual group of characters of G, Vχis the χ-eigenspace of G( for each v∈Vχ,ρ(g)(v) =χ(g)v), and Xis the set of characters χsuch that Vχ= 0. Proposition 21 Assume that an abelian group G acts on a complex torus X := V/. Then the isomorphism class of the group of affine transformations which are lifts of transformations in G determines the real affine type of the action of on V=( ⊗R), in particular the above exact sequence determines the action of G up to a real affine isomorphism of A =( ⊗R)/. Proof By Proposition 17 we may reduce ourselves to the case where Gcontains no translations. Step (I): let us first treat the case where Gis cyclic, generated by an element gof order m. Then we claim that the power gmof a lift gof gdetermines the affine type of the transformation g. In fact, if g(v) =αv +b,Vsplits according to the eigenspaces of the linear map α, which has order exactly m, and we can write V=V1⊕V2, where V1=ker(α −Id). Then g(v1+v2)=(v1+b1)+(α2(v2)+b2), and choosing as the origin in the affine space V2the unique fixed point of g2:= v2→ α2(v2)+b2, we reach the normal form g(v1+v2)=(v1+b1)+α2(v2).Now,gm(v1+v2)=(v1+mb1)+v2, hence gmdetermines b1, and therefore also the normal form of g. Step (II): split V=⊕ χVχaccording the the characters of G, and choose, for each χ, an element gχwhose image under χgenerates χ(G)⊂C∗. For each χ, as in step I, choose as origin in the affine space Vχa fixed point for gχ. Let now gbe an arbitrary element in G: then gacts on Vχby v→ χ(g)v +bχ. We only need to show that bχis uniquely determined. This is clear, by step I, if χ(g)=1. Else, there is an integer rsuch that χ(ggr χ)=1, hence the affine action of ggr χon Vχis uniquely determined, therefore also the one of g. Remark 22 If Gis a general group, we can write V=⊕ ρVρwhere Vρis irreducible (but the same irreducible representation may occur more than once) and let Hρbe a subgroup such that ρ(Hρ)=ρ(G). Argueing as in step II above, the theorem holds for Gif one can prove that the affine action of Hρon Vρis uniquely determined up to affine conjugation. I.e., it suffices to prove the result when Vis irreducible and G⊂Aut (V). Remark 23 In the papers [35,37] we claimed the validity of Proposition 21 for any finite group G, but the proof was incorrect. Does the result hold for any finite group G, at least in the case of an Abelian variety A? We leave this question open, hoping to return on it. In the case of generalized hyperelliptic varieties, we require the condition that the action of Gis free: this implies the following condition (1)∀g∈G,∃χ∈X,χ(g)=1,⇔∪ χ∈Xker(χ ) =G, 123
Topological methods in moduli theory 313 while we may assume that Gcontains no translations, i.e., that ρis injective (equiva- lently , Xspans G∨). With the above in mind, we pass to investigate in more detail the case where G is cyclic, the case where the quotient variety shall be called a Bagnera–de Franchis variety. 5.4 Bagnera–de Franchis varieties of small dimension In view of the characterization given in Proposition 16, we can see a Bagnera–de Franchis variety as the quotient of one of product type, since the actions of Tand g commute (by the property that α2◦φ=φ). Dealing with appropriate choices of Tis the easy part, since, as we saw, the points t2of T2satisfy the property α2(t2)=t2. It suffices to choose T2⊂A2[∗] := ker(α2− IdA2), which is a finite subgroup of A2, and then to pick an isomorphism ψ:T2→ T1⊂A1, such that T1:= Im(ψ) ∩β1={0}. Let us then restrict ourselves to consider Bagnera–de Franchis varieties of product type. We show now how to further reduce to the investigation of primary Bagnera–de Franchis varieties. In fact, in the case of a BdF variety of product type, 2is a G-module, hence a module over the group ring R:= R(m):= Z[G]∼ =Z[x]/(xm−1). This ring is in general far from being an integral domain, since indeed it can be written as a direct sum of cyclotomic rings, which are the integral domains defined as Rm:= Z[x]/Pm(x). Here Pm(x)is the m-th cyclotomic polynomial Pm(x)=0<j<m,(m,j)=1(x−j), where =exp(2πi/m). Then R(m)=⊕ k|mRk. The following elementary lemma, together with the splitting of the vector space Vas a direct sum of eigenspaces for g, yields a decomposition of A2as a direct product A2=⊕ k|mA2,kof G-invariant Abelian subvarieties A2,kon which gacts with eigenvalues of order precisely k. Lemma 24 Assume that M is a module over a ring R =⊕ kRk. Then M splits uniquely as a direct sum M =⊕ kMksuch that Mkis an Rk-module, and the R-module structure is obtained through the projection R →Rk. 123
314 F. Catanese Proof We can write the identity in Ras a sum of idempotents 1 =kek, where ekis the identity of Rk, and ekej=0for j= k. Then each element w∈Mcan be written as w=1w=(kek)w =kekw=: kwk. Hence Mkis defined as ekM. In the situation where we have a primary Bagnera–de Franchis variety 2is a module over the integral domain R:= Rm:= Z[x]/Pm(x), Since 2is a projective module, a classical result (see [290], lemmas 1.5 and 1.6) is that 2splits as the direct sum 2=Rr⊕Iof a free module with an ideal I⊂R, and it is indeed free if the class number h(R)=1 (to see for which integers mthis occurs, see the table in [370], p. 353). To give a complex structure to A2:= (2⊗ZR)/2 it suffices to give a decomposition 2⊗ZC=V⊕¯ V,such that the action of xis holomorphic, which is equivalent to asking that Vis a direct sum of eigenspaces Vλ, for λ=ja primitive m-th root of unity. Writing U:= 2⊗ZC=⊕Uλ, the desired decomposition is obtained by choosing, for each eigenvalue λ, a decomposition Uλ=U1,0 λ⊕U0,1 λsuch that U1,0 λ=U0,1 ¯ λ. The simplest case (see [89] for more details) is the one where I=0,r=1, hence dim(Uλ)=1, so that we have only a finite number of complex structures, depending on the choice of the ϕ(m)/2 indices jsuch that Uj=U1,0 j. The above discussion does however leave open the following question (see [265] for some partial results), which should have an affirmative answer (in the sense that each family of complex structures should contain some Abelian variety). Question: when is then the complex torus A2an Abelian variety? Observe that the classification in small dimension is possible thanks to the obser- vation that the Z-rank of R(or of any ideal I⊂R) cannot exceed the real dimension of A2:inotherwordswehave ϕ(m)≤2(n−1), where ϕ(m)is the Euler function, which is multiplicative for relatively prime numbers, and satisfies ϕ(pr)=(p−1)pr−1when pis a prime number. For instance, when n≤3, then ϕ(m)≤4, and ϕ(pr)≤4iffp=3,5,r=1, or p=2,r≤3. Observe that the case m=2 was completely described, in view of Proposition 16, hence we may assume that m≥3 and ask for the classification for the Bagnera–de Franchis varieties (or manifolds) of order m. Assuming then m≥3, if n≤3, it is only possible to have m=3,4,6,ϕ(m)=2, or m=5,8,10,12,ϕ(m)=4. The classification is then made easier by the fact that, in the above range for m,Ris a P.I.D., hence every torsion free module is free. In particular 2is a free R-module. The classification for n=4, since we must have ϕ(m)≤6, is going to include also the case m=7,9. 123
Topological methods in moduli theory 315 We are not aware of literature dedicated to a precise classification of Bagnera–de Franchis varieties, or generalised hyperelliptic varieties, at least in dimension >3 (see however [264,357] for results in dimension 3). Observe that the hypothesis that Gis a finite group allows to find a G-invariant Hermitian metric on V, hence the affine group extension of by Gis a Bieberbach group, and in each dimension we have only a finite number of those. We end this section mentioning some elementary results which are useful to locate the BdF varieties in the classification theory of algebraic varieties. Proposition 25 The Albanese variety of a Bagnera–de Franchis variety X =A/Gis the quotient A1/(T1+β1). Proof Observe that the Albanese variety H0(1 X)∨/Im(H1(X,Z)) of X=A/Gis a quotient of the vector space V1by the image of the fundamental group of X(actually of its abelianization, the first homology group H1(X,Z)): since the dual of V1is the space of G-invariant forms on A,H0(1 A)G∼ =H0(1 X). We also observe that there is a well defined map X→A1/(T1+β1), since T1is the first projection of T. The image of the fundamental group of Xcontains the image of , which is precisely the extension of 1by the image of T, namely T1. Since we have the exact sequence 1→=π1(A)→π1(X)→G→1 the image of the fundamental group of Xis generated by the image of and by the image of the transformation g, which however acts on A1by translation by β1= [b1]. Remark 26 Unlike the case of complex dimension n=2, there are Bagnera–de Fran- chis varieties X=A/Gwith trivial canonical divisor, for instance some examples are given by: (1) any BdF variety which is standard (i.e., m=2) and is such that A2has even dimension has trivial canonical divisor, as well as (2) some BdF varieties X=A/G=(A1×A2)/(T×G)whose associated product type BdF variety Y=(A1×A2)/Gis primary and elementary (meaning that 2∼ =Rm). For instance, when m=pis an odd prime, there is a primary and elementary BdF variety with trivial canonical divisor if and only if, given the set of integers {1,2,...,[p−1 2]}, there is a partition of it into two sets whose respective sums are the same: for m=7, it suffices to choose the partition {1,2}∪{3}. This partition yields the sets {1,2,4}and {3,5,6}, which determine V=H1,0 as V=H1,0:= U1⊕U2⊕U4. Whereas, for m=17, there are many choices, consisting of the union of two doubletons of the form {j,9−j}. We do not further consider the question of determining exactly the case where the divisor KXis trivial, also because this question can be asked also for the more general case where the action of Gisnotfree(see[316] for a classification in dimension n=3). 123
316 F. Catanese 6 Orbifold fundamental groups and rational K(π, 1)’s 6.1 Orbifold fundamental group of an action In the previous section we have considered quotients X=A/Gof a complex torus A=V/ by the free action of a finite group G. In this case the affine group fitting into the exact sequence 1→=π1(A)→→G→1 equals π1(X). We have also seen that we have the same exact sequence in the more general case where the action of Gis no longer free, and that the group determines in general the affine action of on the real affine space V(resp. on the rational affine space ⊗Q) and that, once the Hodge type of Vis fixed, these varieties are parametrized by a connected complex manifold. In the case where the action of Gis no longer free, we would like to remember the group , in view of its importance, even if it is no longer a ‘bona fide’ fundamental group. This can be done through a more general correspondence which associates to the pair of Xand the group action of Ga group which is called the orbifold fundamental group and can be defined in many ways (see [96,135]). Here is one. Definition 27 Let Zbe a ‘good’ topological space, i.e., arcwise connected and semi- locally 1-connected, so that there exists the universal cover Dof Z. Then we have Z=D/π where π:= π1(Z); denote p:D→Zthe quotient projection. Assume now that a group Gacts properly discontinuously on Z, and set X:= Z/G. Then we define the orbifold fundamental group of the quotient of Zby Gas the group of all the possible liftings of the action of Gon D, more precisely: πor b 1(Z,G):= {γ:D→D|∃g∈G,s.t.p◦γ=g◦p} Remark 28 (1) We obtain an exact sequence, called orbifold fundamental group exact sequence, (∗∗∗)1→π1(Z)→πor b 1(Z,G)→G→1 since for each g∈Gwe have a lifting γof g, because Dis the universal cover, so its associated subgroup is the identity, hence the lifting property is ensured. Moreover the lifting is uniquely determined, given base points z0∈Z,y0∈D, with p(y0)=z0, by the choice of y∈p−1({z0})with γ(y0)=y. (2) In the case where the action of Gis free, then πor b 1(Z,G)=π1(Z/G)=π1(X). (3) If Gacts properly discontinuously on Z, then := πor b 1(Z,G)acts also properly discontinuously on D: in fact the defining property that, for each compact K⊂D,(K,K):= {γ|γ(K)∩K=∅}is finite, follows since p(K)is compact, and since πacts properly discontinuously. 123
Topological methods in moduli theory 317 (4) If the space Zis a K(π, 1), i.e., a classifying space Z=Bπ, then (***) determines the topological type of the G-action. The group acts by conjugation on π, and this yields an homomorphism G→Out(π ) =Aut(π )/ Inn(π), and we just have to recall that the topological action of γ∈is determined by its action on π, takenuptoInn(π ) if we do not keep track of the base point. In fact there is a bijection between homotopy classes of self maps of Zand homo- morphisms of π, taken of course up to inner conjugation (inner conjugation is the effect of changing the base point, and if we do not insist on taking pointed spaces, i.e., pairs (Z,z0)the action of a continuous map on the fundamental group is only determined up to inner conjugation). Clearly a homeomorphism ϕ:Z→Zyields then an associated element π1(ϕ) ∈Out(π ). (5) Even if not useful for computations, one can still interpret the exact sequence (***) as an exact sequence for the fundamental groups of a tower of covering spaces, using the standard construction for equivariant cohomology that we shall discuss later. Just let BG =EG/Gbe a classifying space, and consider the free product action of Gon Z×EG. Then we have a tower of covering spaces (D×EG)→(Z×EG)→(Z×EG)/G, where D×EG is simply connected, and is the universal cover of Y:= (Z×EG)/G, such that π1(Y)∼ =πor b 1(Z,G). Of course, the above definitions seem apparently unrelated to the fundamental group of X, however we can take the maximal open setU⊂Dwhere := πor b 1(Z,G)acts freely. Indeed U=p−1(U), where Uis the open set U:= Z\{z|Gz:= Stab(z)= {IdG}}. The conclusion is that U/ =U/G=: U ⊂X, hence we get exact sequences 1→π1(U)→π1(U)→G→1, 1→π1(U)→π1(U)→→1. We make now the hypothesis that D, hence also Z,Xare normal complex spaces, in particular they are locally contractible, and that the actions are holomorphic: then we have surjections π1(U)→π1(X), π1(U)→π1(Z), moreover π1(U)→π1(D)={1}, and finally we get π1(Z)=π1(U)/π1(U), πor b 1(Z,G)→π1(X)→0. Example 29 Assume that Z, X are smooth complex curves Z=C,X=C. Then for each point pi∈X\U we get the conjugate γi∈π1(U)of a circle around the point pi, and we have ker(π1(U)→π1(X)) = {γi}, i.e., the kernel is normally generated by these loops. 123
318 F. Catanese Each of these loops γimaps to an element ci∈G, and we denote by mithe order of the element (ci)in the group G. Then γmi i∈π1(U), and it is the conjugate of small circle around a point of Z\U, hence it maps to zero in π1(Z). One can see that in this case πor b 1(Z,G)=π1(U)/γmi i and the kernel of the natural surjection onto π1(X)is normally generated by the γi’s. This means that, if the genus of X=Cis equal to g, then the orbifold fundamental group is isomorphic to the abstract group πg,m1,..., md:= α1,β 1,...,α g,β g,γ 1,...,γ d|d 1γjg 1[αi,β i]=1, γm1 1=···=γmd d=1. And the orbifold exact sequence is an extension, called Nielsen extension, of type (NE)1→πg→πg,m1,..., md→G→1. A similar description holds in general (see [96], definition 4.4 and proposition 4.5, pp. 25–26 ), at least when Zis a complex manifold. Remark 30 The notion of orbifold fundamental group has turned out to be quite useful also in real algebraic geometry, where one considers the action of complex conjugation on a real algebraic variety (see [97] for example). 6.2 Rational K(π, 1)’s: basic examples An important role is also played by complex Rational K(π , 1)’s, i.e., quasi projective varieties (or complex spaces) Zsuch that Z=D/π, where Dis a contractible manifold (or complex space) and the action of πon Dis properly discontinuous but not necessarily free. While for a K(π , 1)we have H∗(π, Z)∼ =H∗(Z,Z),H∗(π, Z)∼ =H∗(Z,Z),for a rational K(π , 1), as a consequence of Proposition 46 and of the spectral sequence of Theorem 62,wehaveH∗(π, Q)∼ =H∗(Z,Q)and therefore also H∗(π, Q)∼ = H∗(Z,Q). Typical examples of such rational K(π, 1)’s are: (1) quotients of a bounded symmetric domain Dby a subgroup ⊂Aut(D)which is acting properly discontinuously (equivalently, is discrete); especially note- worthy are the case where is cocompact, meaning that X=D/ is compact, and the finite volume case where the volume of Xvia the invariant volume form for Dis finite. 123
Topological methods in moduli theory 319 (2) the moduli space of principally polarized Abelian Varieties, which is an important special case of the above examples; (3) the moduli space of curves Mg. For the first example the property is clear (we observed already that such a domain D is contractible). For the second example it suffices to observe that the moduli space of Abelian Varieties of dimension g, and with a polarization of type (d1,d2,...,dg)is the quotient of Siegel’s upper half space Hg:= τ∈Mat(g,g,C)|τ=tτ,Im(τ ) > 0, which is biholomorphic to a bounded symmetric domain of type III in E. Cartan’s classification [77], by the properly discontinuous action of the group Sp(D,Z):= M∈Mat(2g,Z)|tMDM =D, where D:= diag(d1,d2,...,dg), D:= 0D −D0 and, M:= αβ γδ ,τ→−D(Dα−τγ) −1(Dβ−τδ). In fact, these Abelian varieties are quotients Cg/, where is generated by the columns of the matrices D,−τ, and the Hermitian form associated to the real matrix !m(τ )−1has imaginary part which takes integral values on ×, and its associated antisymmetric matrix in the chosen basis is equal to D. The quotient Ag,D:= Hg/Sp(D,Z)is not compact, but of finite volume, and there exist several compactifications which are projective algebraic varieties (see [163,176, 311]). 6.3 The moduli space of curves The most useful (and first fully successful) approach to the moduli space of curves of genus gis to view it as a quotient (∗∗)Mg=Tg/Mapg of a connected complex manifold Tgof dimension 3g−3+a(g), called Teichmüller space, by the properly discontinuous action of the Mapping class group Mapg(here a(0)=3,a(1)=1,a(g)=0,∀g≥2 is the complex dimension of the group of automorphisms of a curve of genus g). A key result (see [217,237,356]) is the 123
320 F. Catanese Theorem 31 Teichmüller space Tgis diffeomorphic to a ball, and the action of Mapg is properly discontinuous. Denoting as usual by πgthe fundamental group of a compact complex curve C of genus g, we have in fact a more concrete description of the mapping class group (see [9,10,44,45,125,165,212,220] for general results on braid and mapping class groups): (M)Mapg∼ =Out+(πg). The above superscript +refers to the orientation preserving property. There is a simple algebraic way to describe the orientation preserving property: any automorphism of a group induces an automorphism of its abelianization, and any inner automorphism acts trivially. Hence Out(G)acts on Gab, in our particular case Out(πg)acts on πab g∼ =Z2g, and Out+(πg)⊂Out(πg)is the inverse image of SL(2g,Z). The above isomorphism (M) is of course related to the fact that Cis a K(πg,1),as soon as g≥1. As we already discussed, there is a bijection between homotopy classes of self maps of Cand endomorphisms of πg, taken up to inner conjugation. Clearly a homeomor- phism ϕ:C→Cyields then an associated element π1(ϕ) ∈Out(πg). Such a homeomorphism acts then on the second homology group H2(C,Z)∼ = Z[C], where the generator [C]corresponds to the orientation associated to the complex structure; the condition H2(ϕ) =+1 that ϕis orientation-preserving translates into the above algebraic condition on ψ:= π1(ϕ). That the induced action ψab on the Abelianization π1(C)ab ∼ =πab g∼ =Z2gsatisfies: ∧2g(ψab)acts as the identity on ∧2g(Z2g)∼ =Z. In other words, the image of the product of commutators := j[αj,βj]is sent to a conjugate of . Turning now to the definition of the Teichmüller space Tg, we observe that it is somehow conceptually easier to give the definition of Teichmüller space for more general manifolds. 6.4 Teichmüller space Let Mbe an oriented real differentiable manifold of real dimension 2n, for simplicity let’s assume that Mis compact. Ehresmann [149] defined an almost complex structure on Mas the structure of a complex vector bundle on the real tangent bundle TM R: namely, the action of √−1 on TM Ris provided by an endomorphism J:TM R→TM R,with J2=−Id. Equivalently, as done for the complex tori, one gives the decomposition of the com- plexified tangent bundle TM C:= TM R⊗RCas the direct sum of the i, respectively −ieigenbundles: TM C=TM 1,0⊕TM 0,1where TM 0,1=TM 1,0. 123
Topological methods in moduli theory 321 The space AC(M)of almost complex structures, once TM R(hence all associ- ated bundles) is endowed with a Riemannian metric, is a subset of the Fréchet space H0(M,C∞(End(TM R))) . A closed subspace of AC(M)consists of the set C(M)of complex structures: these are the almost complex structures for which there are at each point xlocal holomorphic coordinates, i.e., C-valued functions z1,...,znwhose differentials span the dual (TM 1,0 y)∨of TM 1,0 yfor each point yin a neighbourhood of x. In general, the splitting TM ∨ C=(TM 1,0)∨⊕(TM 0,1)∨ yields a decomposition of exterior differentiation of functions as df =∂f+¯ ∂f, and a function is said to be holomorphic if its differential is complex linear, i.e., ¯ ∂f=0. This decomposition d=∂+¯ ∂extends to higher degree differential forms. The theorem of Newlander and Nirenberg [312], first proven by Eckmann and Frölicher in the real analytic case [144] characterizes the complex structures through an explicit equation: Theorem 32 (Newlander–Nirenberg) An almost complex structure J yields the struc- ture of a complex manifold if and only if it is integrable, which means ¯ ∂2=0. Obviously the group of oriented diffeomorphisms of Macts on the space of complex structures, hence one can define in few words some basic concepts. Definition 33 Let Dif f+(M)be the group of orientation preserving diffeomorphisms of M, and let C(M)be the space of complex structures on M.LetDiff0(M)⊂ Diff+(M)be the connected component of the identity, the so called subgroup of diffeomorphisms which are isotopic to the identity. Then Dehn [132] defined the mapping class group of Mas Map(M):= Diff+(M)/Dif f0(M), while the Teichmüller space of M, respectively the moduli space of complex structures on Mare defined as T(M):= C(M)/Diff0(M), M(M):= C(M)/Diff+(M). From these definitions follows that M(M)=T(M)/Map(M). The simplest examples here are two: complex tori and compact complex curves. In the case of tori a connected component of Teichmüller space (see [100] and also [103]) is an open set Tnof the complex Grassmann Manifold Gr(n,2n), image of the open set of matrices ∈Mat(2n,n;C)|indet() > 0. 123
322 F. Catanese This parametrization is very explicit: if we consider a fixed lattice ∼ =Z2n, to each matrix as above we associate the subspace of C2n∼ =⊗Cgiven as V=Cn, so that V∈Gr(n,2n)and ⊗C∼ =V⊕¯ V. Finally, to we associate the torus YV:= V/pV() =( ⊗C)/( ⊕¯ V), pV:V⊕¯ V→Vbeing the projection onto the first addendum. It was observed however by Kodaira and Spencer already in their first article on deformation theory ([248], and volume II of Kodaira’s collected works) that the map- ping class group SL(2n,Z)does not act properly discontinuously on Tn. The case of compact complex curves Cis instead the one which was originally considered by Teichmüller. In this case, if the genus gis at least 2, the Teichmüller space Tgis a bounded domain, diffeomorphic to a ball, contained in the vector space of quadratic differentials H0(C,OC(2KC)) ∼ =C3g−3on a fixed such curve C. In fact, for each other complex structure on the oriented 2-manifold Munderlying Cwe obtain a complex curve C, and there is a unique extremal quasi-conformal map f:C→C, i.e., a map such that the Beltrami distortion μf:= ¯ ∂f/∂ fhas minimal norm (see for instance [217]or[8]). The fact that the Teichmüller space Tgis diffeomorphic to a ball (see [356]for a simple proof) is responsible for the fact that the moduli space of curves Mgis a rational K(π, 1). 6.5 Singularities of Mg,I Teichmüller’s theorem says that, for g≥2, Tg⊂C3g−3is an open subset diffeomor- phic to a ball. Moreover Mapg∼ =Map(C)acts properly discontinuously on Tg,but not freely. The lack of freeness of this action is responsible of the fact that Mgis a singular complex space (it is quasi-projective by the results of Mumford and Gieseker, see [302,304]). Therefore one has to understand when a class of complex structure Con the man- ifold Mis fixed by an element γ∈Mapg. Since γis represented by the class of an orientation preserving diffeomorphism ϕ:M→M, it is not difficult to see that this situation means that the diffeomorphism ϕcarries the given complex structure to itself, or, in other words, the differential of ϕpreserves the splitting of the complexified tangent bundle of M,TM C=TM 1,0⊕TM 0,1. This is then equivalent to ¯ ∂ϕ =0, i.e., ϕis a biholomorphic map, i.e., ϕ∈Aut(C). Conversely, let γ∈Aut(C)be a nontrivial automorphism. Consider then the graph of γ,γ⊂C×C, and intersect it with the diagonal ⊂C×C: their intersec- 123
Topological methods in moduli theory 323 tion number must be a nonnegative integer, because intersection points of complex subvarieties carry always a positive local intersection multiplicity. This argument was used by Lefschetz to prove the following Lemma 34 (Lefschetz’ lemma) If we have an automorphism γ∈Aut(C)of a pro- jective curve of genus g ≥2, then γis homotopic to the identity iff it is the identity: γ∼ hId C⇔γ=Id C. Proof The self intersection of the diagonal 2equals the degree of the tangent bundle to C, which is 2 −2g.Ifγis homotopic to the identity, then γ·=2−2g<0, and this is only possible if γ=, i.e., iff γ=idC. In particular, since Cis a classifying space Bπgfor the group πg,γ:C→C is such that the homotopy class of γis determined by the conjugacy class of π1(γ ) :π1(C,y0)→π1(C,γ ·y0). Hence we get an injective group homomorphism ρC:Aut(C)→Aut(πg) Inn(πg)=Out(πg). Actually, since a holomorphic map is orientation preserving, we have that ρC: Aut(C)→Out+(πg)=Mapg∼ =Diff+(C) Diff0(C). The conclusion is then that curves with a nontrivial group of automorphisms Aut(C)={Id C}correspond to points of Tgwith a non trivial stabilizer for the action of Mapg. Remark 35 There is of course the possibility that the action of Mapgis not effective, i.e., there is a normal subgroup acting trivially on Tg. This happens exactly in genus g=2, where every curve is hyperelliptic and the normal subgroup of order 2 generated by the hyperelliptic involution is exactly the kernel of the action. The famous theorem of Hurwitz gives a precise upper bound |Aut(C)|≤84(g−1) for the cardinality of the stabilizer Aut(C)of C(that this is finite follows also by the proper discontinuity of the action). Now, coming to Sing(Mg), we get that Mgis locally analytically isomorphic to a quotient ‘singularity’ C3g−3/Aut(C)(indeed, as we shall see, to the quotient singularity of H1(C)/ Aut(C)at the origin). Now, given a quotient singularity Cn/G, where Gis a finite group, then by a well known lemma by H. Cartan [78] we may assume that Gacts linearly, and then a famous theorem by Chevalley [124] and Shephard and Todd [338] says that the quotient Cn/Gis smooth if and only if the action of G⊂GL(n,C)is generated by pseudo-reflections, i.e., matrices which are diagonalizable with (n−1)eigenvalues equal to 1, and the last one a root of unity. Theorem 36 Let G ⊂GL(n,C),letGpr be the normal subgroup of G generated by pseudo reflections, and let Gqbe the factor group. Then 123
324 F. Catanese (1) the quotient Cn/Gpr is smooth, (2) We have a factorization of the quotient map Cn→Cn/GasCn→(Cn/Gpr )∼ = Cn→Cn/Gqand Cn/G is singular if G = Gpr . This result is particularly interesting in this situation, since the only pseudo-reflection occurs for the case of the hyperelliptic involution in genus g=3. Teichmüller theory can be further applied in order to analyse the fixed loci of finite subgroups Gof the mapping class group (see [96,237,356]) Theorem 37 (Refined Nielsen realization) Let G ⊂Mapgbe a finite subgroup. Then Fix(G)⊂Tgis a non empty complex manifold, diffeomorphic to a ball. Remark 38 (1) The name Nielsen realization comes from the fact that Nielsen [313– 315] conjectured that any topological action of a finite group Gon a Riemann surface could be realized as a group of biholomorphic automorphisms if and only if the action would yield an embedding ρ:G→Mapg. (2) The question can be reduced to a question of algebra, by the following obser- vation. The group πg,forg≥2, has trivial centre: in fact, two commuting hyperbolic transformations in PSL(2,R)can be simultaneously written as Möbius transforma- tions of the form z→ λz,z→ μz, hence if the first transformation were in the centre of πg⊂PSL(2,R), the group πgwould be abelian, a contradiction. Then πg∼ =Inn(πg)and from the exact sequence 1→πg∼ =Inn(πg)→Aut(πg)→Out(πg)→1 and an injective homomorphism ρ:G→Mapg=Out+(πg)one can pull-back an extension5 1→πg→ˆ G→G→1. (4) The main question is to show that the group ˆ Gis isomorphic to an orbifold fundamental group of the form πg,m1,...,md. Because then we have a Nielsen extension, and the epimorphism μ:πg,m1,...,md→G→1 combined with Riemann’s existence theorem finds for us a curve Cover which Gacts with the given topological type: such a curve Cis given by the ramified covering of a curve Cof genus g, branched on dpoints y1,...,ydand with monodromy μ. (5) Since πgis torsion free (hyperbolic elements in PSL(2,R)have infinite order), we see that the branch points correspond to conjugacy classes of cyclic subgroups of finite order in ˆ G, and that their order miin ˆ Gequals the order of their image in G. In turn, the question can be reduced to showing that there is a differentiable action of the group Gon the curve Cinducing the topological action ρ. There is the following 5The same result holds without the assumption that ρbe injective, since it suffices to consider the image G=Im(ρ), apply the pull-back construction to get 1 →πg→ˆ G→G→1, and then construct ˆ Gas the inverse image of the diagonal under the epimorphism ˆ G×G→G×G. 123
Topological methods in moduli theory 325 Lemma 39 Given ρ:G→Mapg=Diff+(C)/Dif f0(C), if there is a homomor- phism ψ:G→Diff+(C)whose projection to Mapgis ρ, then there is a complex curve with a G action of topological type ρ. Proof In fact, by Cartan’s lemma [78], at each fixed point x∈C, there are local coordinates such that in these coordinates the action of the stabilizer Gxof xis linear (in particular, if we assume that the action of Gis orientation preserving, then Gx⊂SO(2,R), and Gxis a cyclic group of rotations). Therefore, for each γ∈G,γ= 1G, the set of fixed points Fix(γ ), which is closed, is either discrete (hence finite), or it consists of a discrete set plus a set (the points x∈Fix(γ ) where the derivative is the identity) which is open and closed. Since C is connected, the only possible alternative is that, for γ= 1G, the set of fixed points Fix(γ ) is finite. Now, it is easy to see that the quotient C/G=: Cis a smooth Riemann surface and we can take a complex structure on Cand lift it to C, so that C=C/Gas complex curves. Remark 40 In general Markovi´c has proven that the mapping class group cannot be realized, as hoped by Thurston, as a subgroup of the group of diffeomorphisms of a Riemann surface of genus g. There are other recent results concerning non finite groups (see [41]). Remark 41 (1) It is interesting to observe that one can view the orbifold fundamental group, in the case of curves, as the quotient of a bona fide fundamental group. Let TC be the tangent bundle of the complex curve, and assume that Ghas a differentiable action on C, so that Gacts on TC, preserving the zero section. We can further assume, by averaging a Riemannian metric on TC, that Gpreserves the sphere bundle of TC, SC := {v∈TC||v|=1}. Since each stabilizer Gxis a cyclic group of rotations, the conclusion is that Gacts on SC by a free action, and we get an exact sequence (Sei f ert)1→π1(SC)→π1(SC/G)→G→1. It is well known (see e.g. [105]) that the fundamental group of the fibre bundle SC is a central extension 1→Z→π1(SC)→πg→1, and is the quotient of the direct product Z×F2gby the subgroup normally generated by the relation c2g−2g 1[αi,β i]=1, where cis a generator of the central subgroup c∼ =Z. 123
326 F. Catanese Taking the quotient by the subgroup cgenerated by cone obtains the orbifold group exact sequence. The fibration SC/G→C=C/Gis a so-called Seifert fibration, with fibres all homemorphic to S1, but the fibres over the branch points yj are multiple of multiplicity mj. Now, an injection G→Out+(πg)determines an action of Gon π1(SC)(sending cto itself), an extension of the type we denoted (Seifert), and a topological action on the classifying space SC. (2) The main point of the proof of the Nielsen realization is however based on analysis, and, more precisely, on the construction of a Morse function fon Tgof class C1which is G-invariant, strictly convex and proper, and therefore has only one minimum. The unique minimum must be G-invariant, hence we get a point Cin Fix(G). Since the function is strictly convex and proper, it does not have other critical points, and it has only one minimum: therefore Morse theory tells that Tgis diffeomorphic to a ball. Now, in turn, the locus Fix(G)is a non empty submanifold, and the group Gacts nontrivially on the normal bundle of Fix(G)in Tg. The same analysis applies to the restriction of the Morse function to any connected component Yof Fix(G). The function has exactly one critical point D, which is an absolute minimum on Y, hence Yis also diffeomorphic to a ball. To show that Fix(G)is connected it suffices to show that C=D, i.e., that Dis also a critical point on the whole space Tg. But the derivative of fat Dis a G-invariant linear functional, vanishing on the subspace of G-invariants: therefore the derivative of fat Dis zero. Hence Fix(G)=Y,aswellasTg, is diffeomorphic to a ball. (3) It may happen that Fi x(G)may be also the fixed locus for a bigger subgroup H⊂Mapg. We shall see examples later in Examples 189,190. To determine when this happens is an interesting question, fully answered by Cornalba in the case where Gis a cyclic group of prime order [127] (this result is the key to understanding the structure of the singular locus Sing(Mg)). A partial answer to the question whether Gis not a full subgroup (i.e., ∃H⊃G, Fix(H)=Fix(G)was given by several authors, who clarified the orbifold funda- mental groups associated to the quotients C→C/G→C/H[275,327,343]. 6.6 Group cohomology and equivariant cohomology The roots of group cohomology go back to Jacobi and to the study of periodic meromor- phic functions as quotients of quasi-periodic holomorphic functions; these functions, which can more generally be taken as vector-valued functions s:Cn→Cm,are solutions to the functional equation s(x+γ) =fγ(x)s(x), where ⊂Cnis a discrete subgroup, ∀γ∈⊂Cn,x∈Cn,fγ:Cn→GL(m,C), and clearly fγsatisfies then the cocycle condition 123
Topological methods in moduli theory 327 fγ+γ(x)=fγ(x+γ)fγ(x), ∀γ,γ ∈, ∀x∈Cn. Later on, the ideas of Jacobi were generalised through the concept of vector bundles, since the cocycle fγ(x)can be used to constructing a vector bundle on the quotient manifold Cn/ taking a quotient of the trivial bundle Cn×Cm→Cnby the relation (x,v) ∼(x+γ, fγ(x)v ), and the cocycle relation just says that the above is an equivalence relation. The main difficulty classically consisted in simplifying as much as possible the form of these cocycles, replacing fγ(x)by an equivalent one φ(x+γ)fγ(x)φ(x)−1,forφ:Cn→ GL(m,C). The construction can be vastly generalized by viewing any bundle on a classifying space Y=B, quotient of a contractible space X=E, as the quotient of the trivial bundle X×Fby an action which covers the given action of Gon X, hence such that there exists fγ:X→Homeo(F)with γ(x,v)=(γ (x), fγ(x)(v)) ⇒∀γ,γ ,∀x∈X,fγγ(x)=fγ(γ (x)) fγ(x). As remarked by David Mumford [309], classical mathematics led to concrete under- standing of 1-cocycles and 2-cocycles, hence of first and second cohomology groups. But the general machinery of higher cohomology groups is harder to understand con- cretely. As always in mathematics, good general definitions help to understand what one is doing, but explicit calculations remain often a hard task. Let us concentrate first on the special case where we look at the singular cohomology groups of a classifying space. In the case of a classifying space B, we know that for each ring of coefficients R, H∗(, R)=H∗(B, R), and for this it suffices indeed that H∗(, Z)=H∗(B, Z). Which side of the equation is the easier one to get hold of? The answer depends of course on our knowledge of the group and of the classifying space B, which is determined only up to homotopy equivalence. For instance, we saw that an incarnation of B(Z/2)is given by the inductive limit P∞ R:= limn→∞ Pn R. A topologist would just observe that P∞ Ris a CW-complex obtained adding a cell σnin each dimension n, and with boundary map ∂(σ2n)=2σ2n−1,∂(σ 2n−1)=0·σ2n−2,∀n≥1. Therefore H2i(Z/2,Z)=0,H2i−1(Z/2,Z)=Z/2,∀i≥1,and taking the dual complex of cellular cochains one gets: 123
328 F. Catanese H2i(Z/2,Z)=Z/2,H2i−1(Z/2,Z)=0,∀i≥1. There are two important ways how this special calculation generalizes: (i) a completely general construction of a CW complex which is a classifying space Bfor a finitely presented group , (ii) a completely algebraic definition of group cohomology. (i): In fact, assume we are given a finite presentation of as =x1,...,xn|R1(x), . . . Rm(x), which means that is isomorphic to the quotient of the free group Fnwith generators x1,...,xnby the minimal normal subgroup Rcontaining the words Rj∈Fn(Ris called the subgroup of relations, and it is said to be normally generated by the relations Rj). Then the standard construction of the 2-skeleton of Bis the CW-complex B2, of dimension two, which is obtained by attaching, to a bouquet of ncircles (which correspond to the generators x1,...,xn), m2-cells whose respective boundary is the closed path corresponding to the word Rj,∀j=1,...,m. For instance, Z/2=Z/(2Z)=x1|x2 1, and the above procedure produces P2 Ras 2-skeleton. However, the universal cover of P2 Ris the sphere S2, which has a homotopy group π2(S2)∼ =Z, and we know that the higher (i≥2) homotopy groups πi(X)of a space equal the ones of its universal cover. Then a 3-cell is attached to P2 R, obtaining P3 R, and one continues to attach cells in order to kill all the homotopy groups. The same is done more generally to obtain Bfrom the CW-complex B2: 3-cells are attached in order to kill the second homotopy groups, and one then obtains B3; one obtains then B4adding 4-cells in order to kill π3(B3), and so on. The disadvantage of this construction is that from the third step onwards it is no longer so explicit, since calculating homotopy groups is difficult. An easy but important remark used in the proof of a theorem of Gromov (see [91,196]) is that when the group πhas few relations, which more precisely means here n≥m+2, then necessarily there is an integer bwith 0 ≤b≤msuch that the rank of H1(Bπ, Z)equals n−m+b, while rank(H2(Bπ, Z)) ≤b:since by the above construction H2(Bπ2,Z)is the kernel of a linear map ∂:Zm→H1(Bπ1,Z)=Zn, whose rank is denoted by m−b, and we have a surjection H2(Bπ2,Z)→H2(Bπ, Z). Hence, if we consider the cup product bilinear map ∪:H1(Bπ, Z)×H1(Bπ, Z)→H2(Bπ, Z), for each ψ∈H1(Bπ, Z)the linear map φ→ ψ∪φhas a kernel of rank ≥n−m≥2. This means that each element ψ∈H1(Bπ, Z)is contained in an isotropic subspace (for the cup product map) of rank at least 2. 123
Topological methods in moduli theory 329 It follows that if the fundamental group of a space X,:= π1(X)admits a surjection onto π, then the induced classifying continuous map φ:X→Bπ(see 11) has the property •that its induced homomorphism on first cohomology H1(φ) :H1(Bπ, Z)→H1(X,Z) is injective (since H1(φ) :H1(X,Z)→H1(Bπ, Z)is surjective) •the image Wof H1(φ) is such that each element w∈Wis contained in an isotropic subspace of rank ≥2. We shall explain Gromov’s theorem, for the case where Xis a Kähler manifold, in the next section. But let us return to the algebraic point of view (see for instance [223], pp. 355–362, also [63]). (ii): Assume now that Gis a group, so that we have the group algebra A:= Z[G]. Recall the functorial definition of group cohomology and of group homology, which gives a high brow explanation for a rather concrete definition we shall give later. Definition 42 Consider the category of modules Mover the group ring A:= Z[G] (i.e., of abelian groups over which there is an action of G, also called G-modules). Then there are two functors, the functor of invariants M→ MG:= {w∈M|g(w) =w∀g∈G}=∩ g∈Gker(g−Id), associating to each module Mthe submodule of elements which are left fixed by the action of G, and the functor of co-invariants M→ MG:= M/(g∈GIm(g−Id)), associating to each module Mthe minimal quotient module on which the action of G becomes trivial. Both MGand MGare trivial G-modules, and one defines the cohomology groups Hi(G,M)as the derived functors of the functor of invariants, while the homology groups Hi(G,M)are the derived functors of the functor of co-invariants. The relation of these concepts to homological algebra is furnished by the following elementary lemma. Lemma 43 Let Zbe the trivial G -module, i.e., we consider the abelian group Zwith module structure such that every g ∈G acts as the identity. Then we have canonical isomorphisms HomZ[G](Z,M)∼ =MG,M⊗Z[G]Z∼ =MG, given by the homomorphisms f → f(1), respectively w⊗n→[nw]. In particular, we have Hi(G,M)=Exti Z[G](Z,M), Hi(G,M)=TorZ[G] i(Z,M). 123
330 F. Catanese We illustrate now the meaning of the above abstract definition for the case of group cohomology. One can construct an explicit free resolution (a projective resolution would indeed suffice) for the trivial A-module M=Z ···Ln→Ln−1→···→ L1→L0→Z→0. Then for each A-module Mthe cohomology groups Hn(G,M)are computed as the cohomology groups of the complex Hom(L0,M)→Hom(L1,M)→···→Hom(Ln−1,M)→Hom(Ln,M)→···. While the homology groups are calculated as the homology groups of the complex ···→M⊗Z[G]Li+1→M⊗Z[G]Li→M⊗Z[G]Li−1→···. For instance, if Gis a cyclic group of order m, then A=Z[x]/(xm−1)and the free resolution of Zis given by free modules of rank one: the homomorphisms are the augmentation :L0→Z(defined by setting (gag·g):= gag) and the scalar multiplications: (x−1):L2n+1→L2n,(1+x···+xm−1):L2n+2→L2n+1, in view of the fact that (x−1)(1+x···+xm−1)=xm−1≡0∈A. The case m=2 we saw before is a very special case, since then A=Z⊕Zx, with x2=1, hence on the trivial module Z=HomZ[G](Z[G],Z)(1−x)acts as multiplication by 0, 1 +xacts as multiplication by 2. But the same pattern happens for all m:(1−x)acts as multiplication by 0, 1 +x···+xm−1acts as multiplication by m. In particular, the homology groups can be calculated as H2i(Z/m,Z)=0,i≥1,H2i+1(Z/m,Z)∼ =Z/m,i≥0. For more general groups there is a general complex, called the bar-complex, which yields a resolution of the A-module Z. It is easier to first give the concrete definition of the cohomology groups (see also [180]). Definition 44 GivenagroupGand a A:= Z[G]-module M, one defines the group of i-cochains with values in Mas: Ci(G,M):= { f:Gi+1→M|f(γ g0,...,γgi)=γf(g0,...,gi)∀γ∈G}, and the differential di:Ci(G,M)→Ci+1(G,M)through the familiar formula df(g0,...,gi+1):= i+1 0(−1)jf(g0,..., ˆgj,...,gi+1). 123
Topological methods in moduli theory 331 Then the groups Hi(G,M)are defined as the cohomology groups of the complex of cochains, that is, Hi(G,M)=ker(di)/ Im(di−1). Remark 45 (1) In group theory one gives a different, but equivalent formula, obtained by considering a cochain as a function of iinstead of i+1 variables, as follows: ϕ(g1,...,gi):= f(1,g1,g1g2,...,g1g2...gi). One observes in fact that, in view of equivariance of fwith respect to left translation on G,giving fis equivalent to giving ϕ. Then the formula for the differential becomes asymmetrical dϕ(g1,...,gi+1)=g1ϕ(g2,...,gi+1)−ϕ(g1g2,g3,...,gi+1) +ϕ(g1,g2g3,...,gi+1)...(−1)iϕ(g1,g2,g3,...,gi). (2) The second formula is more reminiscent of the classical formulae, and indeed, for i=0, yields H0(G,M)={x∈M|gx −x=0∀g∈G}=MG, Z1(G,M)={ϕ=ϕ(g)|g1ϕ(g2)−ϕ(g1g2)+ϕ(g1)=0∀g1,g2∈G}, B1(G,M)={ϕ=ϕ(g)|∃x∈M,ϕ(g) =gx −x},H1(G,M)=Z1(G,M)/B1(G,M). (3) Hence, if Mis a trivial G-module, then B1(G,M)=0 and H1(G,M)= Hom(G,M). We have been mainly interested in the case of Z-coefficients, however if we look at coefficients in a field k,wehave: Proposition 46 Let G be a finite group, and let k be a field such that the characteristic of k does not divide |G|. Then Hi(G,k)=0∀i≥1. Proof If Mis a k[G]-module, we observe first that MG∼ =HomZ[G](Z,M)∼ =Homk[G](k,M), and then that the functor M→ MGis exact, because MGis a direct summand of M (averaging v→ 1 |G|g∈Ggvyields a projector with image MG). Hence Hi(G,M)= 0∀i≥1. 123
332 F. Catanese 6.7 Group homology, Hopf’s theorem, Schur multipliers To give some explicit formula in order to calculate homology groups, we need to describe the bar-complex, which gives a resolution of the trivial module Z. In order to relate it to the previous formulae for group cohomology, we should preliminary observe that the group algebra Z[G]can be thought of as the subalgebra of the space of functions a:G→Zgenerated by the characteristic functions of elements of G(so that g∈Z[G]is the function such that g(x)=1forx=g,else g(x)=0,x= g. As well known, multiplication on the group algebra corresponds to convolution of the corresponding functions, f1∗f2(x)=Gf1(xy−1)f2(y)dy. It is then clear that the tensor product Z[G]⊗ ZMyields a space of M-valued functions on G(all of them, if the group Gis finite), and we get a space of Z-valued functions on the i-th Cartesian product Gi+1by considering the (i+1)-fold tensor product Ci:= Z[G]⊗ ZZ[G]⊗···⊗ZZ[G], with Z[G]-module structure given by g(x0⊗···⊗xi):= g(x0)⊗···⊗xi. Definition 47 The bar-complex of a group Gis the homology complex given by the free Z[G]-modules Ci(a basis is given by {(g1,...,gi):= 1⊗g1⊗···⊗gi}) and with differential di:Ci→Ci−1(obtained by dualizing the previously considered differential for functions), defined by di(g1,g2,...,gi):= g1(g2,...,gi)+i−1 1(−1)j(g1,...gj−1,gjgj+1,...,gi) +(−1)i(g1,g2,...,gi−1). The augmentation map :C0=Z[G]→Zshows then the the bar-complex is a free resolution of the trivial Z[G]-module Z. As a consequence, the homology groups Hn(G,Z)are then the homology groups of the complex (Cn⊗Z[G]Z,dn), and one notices that, since Z[G]⊗Z[G]Z∼ =Z, (observe in fact that g⊗1=1⊗g1=1⊗1), then Cn⊗Z[G]Zis a trivial Z[G]-module and a free Z-module with basis {(g1,...,gn)}. The algebraic definition yields the expected result for i=1. Corollary 48 H1(G,Z)=Gab =G/[G,G],[G,G]being as usual the subgroup generated by commutators. Proof First argument, from algebra: we have ⊕g1,g2∈GZ(g1,g2)→⊕ g∈GZg→Z 123
Topological methods in moduli theory 333 but the second homomorphism is zero, hence H1(G,Z)=(⊕g∈GZg)/g2−g1g2+g1=Gab, since we have then the equivalence relation g1g2≡g1+g2. Second argument, from topology: H1(X,Z)is the abelianization of the fundamental group, so we just apply this to the case of X=BG, since π1(BG)∼ =G. The machinery of algebraic topology is also useful in order to show the following ‘duality’ statement. Theorem 49 If G is a finite group there is an isomorphism between the group H2(G,Z)and the group of ‘Schur multipliers’ H2(G,C∗). Proof Let us consider the exact sequence of groups 0→Z→C→C∗→1, and use that Hi(G,C)=0fori≥1(see[223]). Hence we have an isomorphism H2(G,C∗)∼ =H3(G,Z). Now, by the universal coefficient formula, the torsion subgroup of Hn(X,Z)equals the torsion subgroup of Hn+1(X,Z). Apply this observation to the case X=BG,n= 2, remarking that the groups Hi(G,Z), hence also the groups Hi(G,Z), are torsion abelian groups for i≥1. Indeed, Hi(G,R)=Hi(BG,R)=Hi(EG,R)G=0). The calculation of the above group H2(G,Z)was achieved by Hopf [215], and we shall sketch the underlying topological idea. Theorem 50 (Hopf) Assume that we have a presentation of a group G as the quotient G=F/R of a free group. Then H2(G,Z)=([F,F]∩R)/[F,R]. Proof (Sketch) Proof I (currently fashionable argument) To the exact sequence of groups 1→R→F→G→1 there correspond continuous maps of classifying spaces. These maps are only well defined up to homotopy, and using a small trick we can arrange the corresponding maps to be the standard inclusion of a fibre into a fibre bundle. More precisely, let BG =EG/Gbe a classifying space for G, and BF =EF/F be a classifying space for F: since the group is free, BF shall be a bouquet of circles. Moreover, since Ris a subgroup of F, its classifying space BR is a covering of BF, indeed it is BR =EF/R, and is a CW-complex of dimension 1 (whence, the theorem of Nielsen that a subgroup Rof a free group Fis also free). 123
334 F. Catanese The surjection F→Gyields an action of Fon EG, which restricts to a trivial action of R. We use the construction of equivariant cohomology to obtain a new classifying space BF:= (EF ×EG)/F. The projection of the product (EF ×EG)→EG yields a fibre bundle (EF × EG)/F→EG/Gwith fibre EF/R. Hence an associated sequence of continuous maps BR =EF/R→(EF ×EG)/F=BF→EG/G=BG, associated to the fibre bundle BF→BG and with 1-dimensional fibre BR. We obtain associated maps of first homology groups H1(BR,Z)=R/[R,R]→H1(BF,Z)=F/[F,F]→H1(BG,Z)=G/[G,G], and since BR,BFhave vanishing i-th homology groups for i≥2, application of the Serre spectral sequence for fibre bundles [336] shows that we have the exact sequence 0→H2(G,Z)→H1(BR,Z)F=(R/[R,R])F→F/[F,F]→G/[G,G]→0. In the above exact sequence appears the group of co-invariants (R/[R,R])F, which is the quotient of R/[R,R]by the relations of the form frf−1=r,∀f∈F,r∈R, or, equivalently, by the subgroup [F,R]generated by the commutators frf−1r−1.In other words, (R/[R,R])F=R/[F,R]. Now, from the exact sequences (∗)0→H2(G,Z)→R/[F,R]→K→0 and 0→K→F/[F,F]→G/[G,G]→0, since ker(F→G/[G,G])=R[F,F], we infer that K=(R[F,F])/[F,F]=R/(R∩[F,F]), where the last equality follows from the third isomorphism theorem for groups. Since [F,R]⊂(R∩[F,F]), Hopf’s theorem follows from (*) above. Proof II (direct argument): follows from the following lemma, since the kernel of R/[F,R]→F/[F,F]clearly equals (R∩[F,F])/[F,R]. 123
Topological methods in moduli theory 335 The relation between algebra and geometry in Hopf’s theorem is further explained in lemma 2.5 of [116]. Lemma 51 Let B G be a CW-complex which is a classifying space for the group G=F/R, such that (i) its 1-skeleton B G1has all the 1-cells in bijective correspondence with a set of generators of F , (ii) its 2-skeleton B G2has all the 2-cells in bijective correspondence with a set of generators of R (hence R/[R,R]is isomorphic to the relative homology group H2(BG2,BG1,Z)). Then the following exact sequence of relative homology 0→H2(BG,Z)→H2(BG,BG1,Z)→H1(BG1,Z)→H1(BG,Z)→0 is isomorphic to 0→([F,F]∩R)/[F,R]→R/[F,R]→F/[F,F]→G/[G,G]→0. Proof (Sketch) The main point is to show that the obvious surjection H2(BG2,BG1,Z)→H2(BG,BG1,Z) reads out algebraically as the surjection R/[R,R]→R/[F,R]. In order to explain this, assume that {rj|j∈J}is a system of free generators of R. Now, BG3is obtained attaching 3-cells in order to kill the second homotopy group of BG2which, in turn, by Hurewicz’ theorem, is the second homology group of its universal cover ˜ BG2. But it is then easy to see that generators for H2(˜ BG2,Z)are obtained applying the covering transformations of G=F/Rto the 2-cells corre- sponding to the elements {rj|j∈J}. One can then show that the boundaries of these 3-cells yield relations on R/[R,R]which are exactly those of the form [a,rj],for a∈F. Example 52 For a finite cyclic group G=Z/m,F=Z, hence [F,F]=0, H2(Z/m,Z)=0 and the result fits with previous calculations. The spectral sequence argument is however more powerful, and yields a much more general result (see [372] and references therein). Theorem 53 (Lyndon–Hochshild–Serre spectral sequence) Let H be any group, and R a normal subgroup, and let G =H/R be the factor group. Assume that A is an H -module, so that A Hand AHare G-modules. Then there are converging first quadrant spectral sequences E2 p,q=Hp(G;Hq(R,A)) ⇒Hp+q(H,A) Ep,q 2=Hp(G;Hq(R,A)) ⇒Hp+q(H,A) 123
336 F. Catanese Example 54 Assume that the group His a metacyclic group, i.e., we have an exact sequence 1→R∼ =Z/n→H→G∼ =Z/m→1. As previously remarked, H2(Z/n,Z)=0, H1(Z/n,Z)=Z/n, while H0(Z/n,Z) is the trivial Z/m-module Z. Hence the spectral sequence has many zeros, and reveals an exact sequence Z/m→H1(Z/m,Z/n)→H2(H,Z)→0. Let xbe a generator of R∼ =Z/n, and ya generator of G∼ =Z/m, which we shall identify to one of its lifts to H(the order of ymay be assumed to be equal to mif and only if the exact sequence splits, or, as one says His split metacyclic). Now, conjugation by ysends xto xa, where (a,n)=1, and we have also am≡ 1(mod n). Hence the G=Z/m-structure of R∼ =Z/n, is described by y(b)=ab ∈ Z/n. Tensoring Rwith the bar-complex of Gwe find ⊕g1,g2∈G(Z/n(g1,g2)) →⊕ g∈G(Z/n)g→Z/n where however the last homomorphism is no longer the zero map, and its image is just the image of multiplication by (a−1), i.e., (a−1)(Z/n)=(n,a−1)Z/nZ. In fact Z/n∼ =Z[G]⊗Z[G]Z/n, with yi·1≡ai∈Z/n. Therefore we obtain that H1(G,Z/n)is the quotient of the kernel of multiplication by (a−1), inside the direct sum ⊕g∈G(Z/n)g, by the subgroup generated by g1g2≡g1+g1(1)g2. Writing the elements in Gmultiplicatively, g1=yi,g2=yj, we get the relations yi+j≡yi+aiyj. For i=0 we get thus y0=0, and then inductively we find the equivalent relations yi=(1+···+ai−1)y,∀1≤i≤m−1,0=ym=(1+···+am−1)y. Hence H1(G,Z/n)⊂Z/(n,r), where we set r:= (1+···+am−1)=(am− 1)/(a−1), is the kernel of multiplication by (a−1), hence it is cyclic of order d:= 1 n(n,a−1)(n,r). The conclusion is that H2(H,Z)is a cokernel of Z/m→Z/d. Edmonds shows in [146], with a smart trick, that the map Z/m→Z/dis the zero map when we have a split metacyclic extension, hence H2(H,Z)∼ =Z/din this case. 123
Topological methods in moduli theory 337 We shall show how to make the computation directly using Hopf’s theorem: this has the advantage of writing an explicit generator inside R/[F,R], a fact which shall be proven useful in the sequel. 6.8 Calculating H2(G,Z)via combinatorial group theory We treat again the case of a split metacyclic group H. H=F/R, where Fis a free group on two generators x,y, and the subgroup Rof relations is normally generated by ξ:= xn,η:= ym,ζ:= xayx−1y−1, with (a,n)=1, am≡1(mod n). Since Ris the fundamental group of the Cayley graph of H, whose vertices corre- sponding to the elements of H=xiyj|0≤i≤n−1,0≤j≤m−1, and Ris a free group on mn +1 generators. We use now the Reidemeister-Schreier algorithm (see [276]), which shows that a basis of Ris given by: ηi:= xiymx−i,ξ:= xn,ξ i,j:= xiyjxy−jx−[i+aj], 0≤i≤n−1,1≤j≤m−1, and where [b]∈{0,...,n−1}is the unique positive representative of the residue class of bin Z/n. We want to calculate H2(G,Z)as the kernel of R/[F,R]→K→0, keeping in mind that, since Fab =F/[F,F]is the free abelian group on generators X,Y, then K=ker(Fab →Gab)is generated by the images of ξ,η,ζ in Fab, i.e., nX,mY, (a−1)X. We conclude that Kis a free abelian group with basis (n,a−1)X,mY,and contains the free abelian group with basis nX,mY. Our trick is then to observe that ξ→ nX,η=η0→ mY, hence H2(G,Z)is the kernel of the surjection Q:= (R/[F,R])/(Zξ+Zη)) →K/(Z(nX)⊕Z(mY)) =(n,a−1)Z/nZ→0. This shall simplify our calculations considerably. Observe that R/[F,R], since [R,R]⊂[F,R], is a quotient of Rab.Rab is a free abelian group on generators , i,j, i, which are the respective images of the elements ξ,ξi,j,η i. Thus R/[F,R], is the quotient of Rab by the relations of the form r∼frf−1,r∈R,f∈F. 123
338 F. Catanese Down to earth, each generator is put to be equivalent to its conjugate by x, respec- tively its conjugate by y. Conjugation by xclearly sends ηito ηi+1, for all i<n−1, while xηn−1x−1= xnymx−n=ξηξ−1, thus in the abelianization we get i=, ∀i, and in our further quotient Qthe classes of the i’s are all zero. Conjugation of ξi,jby xmakes ξi,j∼xxiyjxy−jx−[i+aj]x−1=xi+1yjxy−jx−1−[i+aj]=ξi+1,jor ξi+1,jξ−1, hence the classes of i,jand of i+1,jare the same. Therefore the class of i,jis always equal to the one of 0,j. Conjugation by yjof ξ=xnyields that the class of yjxny−jis trivial. But, observing that the class of xn=ξand all its conjugates is trivial, we can consider the exponents of powers of xas just residue classes modulo n, and avoid to use the square brackets. Then yjxny−j=(yjxy−j)n∼(yjxy−jx−aj)(xajyjxy−jx−2aj) ···(x(n−1)ajyjxy−jx−naj), which means that the class of n0,j=0. We denote by Zjthe class of 0,j. Hence nZj=0, and Qis generated by the classes Zj. In the calculation of conjugation by yon 0,jwe observe that ξi,j∼yyjxy−jx−[aj]y−1=(yj+1xy−j−1x−[aj+1])x[aj+1]yx−[aj]y−1. Hence Zj∼yZjy−1=Zj+1xaj+1yx−ajy−1=Zj+1xaj+1yxy−1x−aj−axaj+ayx−aj−1y−1 =Zj+1Zk 1xaj+1+ka yx−aj−ky−1. For k=−ajwe obtained the desired relation Zj=Zj+1Z−aj 1⇔Zj+1=ZjZaj 1, which can be written additively as yi+1≡yi+aiy. The conclusion is that we have a cyclic group whose order is d:= 1 n(n,a−1)(n,r) and with generator n (n,a−1)Z1, where Z1is the class of ξ0,1=yxy−1x−a. 123
Topological methods in moduli theory 339 6.9 Sheaves and cohomology on quotients, linearizations The method of the bar-complex apparently settles the problem of calculating the cohomology groups of a K(G,1)space, and also to calculate the cohomology groups Hn(X,Q)for a rational K(G,1)space X, except that one must have a good knowledge of the group G, for instance a finite presentation does not always suffice. More generally, Grothendieck [199] approached the question of calculating the cohomology of a quotient Y=X/Gwhere Xis not necessarily contractible. Using sheaf theory, one can reach a higher generality, considering G-linearized sheaves, i.e. sheaves Fon Xwith a lifting of the action of Gto F(this is the analogue to taking cohomology groups on non trivial G-modules). Since this concept is very useful in algebraic geometry, and lies at the heart of some important calculations, we shall try now to briefly explain it, following Mumford’s treatment ([303], appendix to section 2, pp. 22–24, section 7, pp. 65–74, section 12, pp. 108–122). The following is the basic example we have in mind, where we consider two for- mally different cases: (1) Xis a projective variety, Gis a finite group acting on Xwith the property that for each point xthe orbit Gx is contained in an affine open subset of X(this property holds if there is an ample divisor Hwhose linear equivalence class is fixed by G), or (2) Xis a complex space, Gacts properly discontinuously on X(in case where the quotient X/Gis a projective variety, we can apply the G.A.G.A. principle by which there is an equivalence of categories between coherent algebraic sheaves and coherent holomorphic sheaves). Assume that we have a vector bundle Von a quotient variety X/G. Denote by p:X→X/Gthe quotient map and take the fibre product p∗(V):= V×X/GX. We consider the action of Gon V×X, which is trivial on the factor V, and the given one on X. In this way the action of Gextends to the pull back p∗(V)=V×X/GX of V, and the sections of Von X/Gare exactly seen to be the G-invariant sections of p∗(V). Now, a main goal is to construct interesting varieties as quotient varieties X/G, and then study line bundles on them; the following result is quite useful for this purpose. Proposition 55 Let Y =X/G be a quotient algebraic variety, p :X→Ythe quotient map. (1) Then there is a functor between (the category of) line bundles Lon Y and (the category of) G-linearized line bundles Lon X, associating to Lits pull back p∗(L). The functor L→ p∗(L)Gis a right inverse to the previous one, and p∗(L)Gis invertible if the action is free, or if both X and Y are smooth.6 (2) Given a line bundle Lon X , it admits a G-linearization if and only if there is a Cartier divisor D on X which is G-invariant and such that L∼ =OX(D)(observe that OX(D)={f∈C(X)|div( f)+D≥0}has an obvious G-linearization). 6One can indeed relax the hypothesis, requiring only that Xis normal and Yis smooth. 123
340 F. Catanese (3) A necessary condition for the existence of a linearization is that ∀g∈G,g∗(L)∼ =L. If this condition holds, defining the Thetagroup of Las: (L,G):= (ψ, g)|g∈G,ψ:g∗(L)∼ =Lisomorphism, there is an exact sequence 1→C∗→(L,G)→G→1. The splittings of the above sequence correspond to the G-linearizations of L. If the sequence splits, the linearizations are a principal homogeneous space over the dual group Hom(G,C∗)=: G∗of G (namely, each linearization is obtained from a fixed one by multiplying with an arbitrary element in G∗). Proof In (1), the case where Gacts freely is trivial in the holomorphic context, and taken care of by proposition 2, p. 70 of [303] in the algebraic context. For (2) and (3) we refer to [302], section 3, chapter 1, and [85], lemmas 4–6. We only provide an argument for the last statement in (1), for which we do not know of a precise reference. The question is local, hence we may assume that Pis a smooth point of X, and that Gis equal to the stabilizer of P, hence it is a finite group. The invertible sheaf is locally isomorphic to OX, except that this isomorphism does not respect the linearization. The G-linearization of Lcorresponds then locally to a character χ:G→C∗. In particular, there exists an integer msuch that the induced linearization on L⊗mcorresponds to the trivial character on OX. Let Rbe the ramification divisor of the quotient map p:X→Y=X/G, and let X0:= X\Sing(R), which is G-invariant. Set Y0:= p(X0). Step I: p∗(L)Gis invertible on Y0. This is clear on the open set which is the complement of the branch set p(R).Atthe points of Rwhich are smooth points, then the Gaction can be linearized locally, and it is a pseudoreflection; so in this case the action of Ginvolves only the last variable, hence the result follows from the dimension 1 case, where every torsion free module is locally free. Step II. Let M:= p∗(L)G. We claim that Mis invertible at every point. Pick a point ynot in Y0and a local holomorphic Stein neighbourhood Uof y, say biholomorphic to a ball. Then U0:= U∩Y0has vanishing homology groups Hi(U0,Z)for i=1,2. By the exponential sequence H1(U0,O∗ Y)∼ =H1(U0,OY), which is a vector space hence has no torsion elements except the trivial one. Since M⊗mis trivial on U0, we conclude that Mis trivial on U0. By Hartogs’ theorem there is then an isomorphism between G-invariant sections of Lon p−1(U)and sections of OYon U. Thus, the question of the existence of a linearization is reduced to the algebraic question of the splitting of the central extension given by the Theta group. This question is addressed by group cohomology theory, as follows (see [223]). 123
Topological methods in moduli theory 341 Proposition 56 An exact sequence of groups 1→M→→G→1, where M is abelian, is called an extension. Conjugation by lifts of elements of G makes M a G-module. Each choice of a lift eg∈for every element g ∈G determines a 2-cocycle ψ(g1,g2):= eg1eg2e−1 g1g2. The cohomology class [ψ]∈H2(G,M)is independent of the choice of lifts, and in this way H2(G,M)is in bijection with the set of strict isomorphism classes of extensions 1→M→→G→1(i.e., one takes isomorphisms inducing the identity on M and G). Whereas isomorphism classes of extensions 1→M→→ G→1(they should only induce the identity on G) are classified by isomorphism classes of pairs (M,ψ). In particular, we have that the extension is a direct product M×Gif and only if Mis a trivial G-module (equivalenty, Mis in the centre of ) and the class [ψ]∈H2(G,M) is trivial. Corollary 57 Let Lbe an invertible sheaf whose class in Pi c(X)is G-invariant. Then the necessary and sufficient condition for the existence of a linearization is the triviality of the extension class [ψ]∈H2(G,C∗)of the Thetagroup (G,L). The group H2(G,C∗)is the group of Schur multipliers (see again [223], p. 369) , which for a finite group is, as we saw in Theorem 49, another incarnation of the second homology group H2(G,Z). It occurs naturally when we have a projective representation of a group G. Since, from a homomorphism G→PGL(r,C)one can pull back the central extension 1→C∗→GL(r,C)→PGL(r,C)→1⇒1→C∗→ˆ G→G→1, and the extension class [ψ]∈H2(G,C∗)is the obstruction to lifting the projective representation to a linear representation G→GL(r,C). It is an important remark that, if the group Gis finite, and n=or d (G), then the cocycles take value in the group of roots of unity μn:= {z∈C∗|zn=1}. Example 58 Let Ebe an elliptic curve, with the point Oas the origin,and let Gbe the group of 2-torsion points G:= E[2]acting by translation on E. The divisor class of 2O is never represented by a G-invariant divisor, since all the G-orbits consist of 4 points, and the degree of 2Ois not divisible by 4. Hence, L:= OE(2O)does not admit a G- linearization. However, we have a projective representation on P1=P(H0(OE(2O)), where each non zero element η1of the group fixes 2 divisors: the sum of the two points corresponding to ±η1/2, and its translate by another point η2∈E[2]. 123
342 F. Catanese The two group generators yield two linear transformations, which act on V= Cx0⊕Cx1as follows: η1(x0)=x1,η 1(x1)=x0,η 2(xj)=(−1)jxj. The linear group generated is however D4, since η1η2(x0)=x1,η 1η2(x1)=−x0. Example 59 The previous example is indeed a special case of the Heisenberg exten- sion, and Vgeneralizes to the Stone von Neumann representation associated to an abelian group G, which is nothing else than the space V:= L2(G,C)of square integrable functions on G(see [221,303]). Gacts on V=L2(G,C)by translation f(x)→ f(x−g),G∗acts by multiplication with the given character f(x)→ f(x)·χ(x)), and the commutator [g,χ]acts by the scalar multiplication with the constant χ(g). The Heisenberg group is the group of automorphisms of Vgenerated by G,G∗and by C∗acting by scalar multiplication, and there is a central extension 1→C∗→Heis(G)→G×G∗→1, whose class is classified by the C∗-valued bilinear form (g,χ) → χ(g), an element of ∧2(Hom(G×G∗,C∗)) ⊂H2(G×G∗,C∗). The relation with Abelian varieties A=V/ is through the Thetagroup associated to an ample divisor L. In fact, since by the theorem of Frobenius the alternating form c1(L)∈H2(A,Z)∼ = ∧2(Hom(, Z)) admits, in a suitable basis for , the normal form D:= 0D −D0 D:= diag(d1,d2,...,dg), d1|d2|...|dg. The key property is that, if one sets G:= Zg/DZg, then Lis invariant exactly for the translations in (G×G∗)∼ =(G×G)⊂A, and the Thetagroup of Lis just isomorphic to the Heisenberg group Heis(G). The nice part of the story is the following useful result, which was used by Atiyah in the case of elliptic curves, to study vector bundles over these [12]. Proposition 60 Let G be a finite abelian group, and let V := L2(G,C)be the Stone- von Neumann representation. Then V ⊗V∨is a representation of (G×G∗)and splits as the direct sum of all the characters of (G×G∗), taken with multiplicity one. 123
Topological methods in moduli theory 343 Proof Since the centre C∗of the Heisenberg group Heis(G)acts trivially on V⊗V∨, V⊗V∨is a representation of (G×G∗). Observe that (G×G∗)is equal to its group of characters, and its cardinality equals to the dimension of V⊗V∨, hence it shall suffice, and it will be useful for applications, to write for each character an explicit eigenvector. We shall use the notation g,h,kfor elements of G, and the notation χ, η,ξ for elements in the dual group. Observe that Vhas two bases, one given by {g∈G}, and the other given by the characters {χ∈G∗}. In fact, the Fourier transform yields an isomorphism of the vector spaces V:= L2(G,C)and W:= L2(G∗,C) F(f):= ˆ f,ˆ f(χ) := f(g)(χ, g)dg. The action of h∈Gon Vsends f(g)→ f(g−h), hence for the characteristic functions in C[G],g→ g+h. While η∈G∗sends f→ f·η, hence χ→ χ+η, where we use the additive notation also for the group of characters. Restricting Vto the finite Heisenberg group which is a central extension of G×G∗ by μn, we get a unitary representation, hence we identify V∨with ¯ V. This said, a basis of V⊗¯ Vis given by the set {g⊗¯χ}. Given a vector g,χ ag,χ(g⊗¯χ)the action by h∈Gsends it to g,χ (χ , h)ag−h,χ(g⊗¯χ), while the one by η∈G∗sends it to g,χ (η, g)ag,χ −η(g⊗¯χ). Hence one verifies right away that Fk,ξ := g,χ (χ −ξ,g−k)(g⊗¯χ) is an eigenvector with eigenvalue (ξ, h)(η, k)for (h,η) ∈(G×G∗). Remark 61 The explicit calculation reproduced above is based on the fact that the Fourier transform does not commute with the action of the Heisenberg group: the action of Gon Vcorresponds to the action of G=(G∗)∗on W, while the action of G∗on Wcorresponds to the multiplication of functions f(g)∈Vwith the conjugate ¯χof the character χ. In the next subsection we shall give an explicit example where the Heisenberg group is used for a geometrical construction. Now let us return to the general discussion of cohomology of G-linearized sheaves. 123
344 F. Catanese In the special case where the action is free there is an isomorphism between the category of G-linearized sheaves Fon X, and sheaves FGon the quotient Y;wehave then Theorem 62 (Grothendieck [199]) Let Y =X/G, where G acts freely on X , and let Fbe a G-linearized sheaf on X , let FGbe the G -invariant direct image on Y. Then there is a spectral sequence converging to a suitable graded quotient of Hp+q(Y,FG)and with E1term equal to H p(G,Hq(X,F)); we write then as a shorthand notation: Hp(G,Hq(X,F)) ⇒Hp+q(Y,FG). The underlying idea is simple, once one knows (see [79,214]) that there is a spectral sequence for the derived functors of a composition of two functors: in fact one takes here the functor F→ H0(X,F)G, which is manifestly the composition of two functors F→ H0(X,F)and A→ AG, but also the composition of two other functors, since H0(X,F)G=H0(Y,FG). The main trouble is that the functor F→ FGis no longer exact if we drop the hypothesis that the action is free. A particular case is of course the one where Fis the constant sheaf Z: in the case where the action is not free one has in the calculation to keep track of stabilizers Gx of points x∈X, or, even better, to keep track of fixed loci of subgroups H⊂G. Since this bookkeeping is in some situations too difficult or sometimes also of not so much use, Borel [56] proposed (in a way which is similar in spirit to the one of Grothendieck) to consider a generalization of the cohomology of the quotient (see [173] for a nice account of the theory and its applications in algebraic geometry). Definition 63 Let Gbe a Lie group acting on a space X, and let BG =EG/Gbe a classifying space for G. Here we do not need to specify whether Gacts on the left or on the right, since for each left action (g,x)→ gx there is the mirror action xg := g−1x. Letting Rbe an arbitrary ring of coefficients, the equivariant cohomology of X with respect to the action of Gis defined in one of the following equivalent ways. (1) Assume that Gacts on both spaces Xand EG from the left: then we have the diagonal action g(e,x):= (ge,gx)and Hi G(X,R):= Hi((EG ×X)/ G,R). (2) (as in [173]) Assume without loss of generality that Gacts on Xfrom the left, and on EG from the right: then Hi G(X,R):= Hi(EG ×GX,R), where the space EG ×GXis defined to be the quotient of the Cartesian product EG ×Xby the relation (e·g,x)∼(e,g·x). 123
Topological methods in moduli theory 345 Remark 64 Observe that, in the special case where Xis a point, and Gis discrete, we reobtain the cohomology group Hi(G,R). Example 65 The consideration of arbitrary Lie groups is rather natural, for instance if G=C∗, then BG =P∞ C=(C∞\{0})/C∗. Similarly the infinite Grassmannian Gr(n,C∞)is a classifying space for GL(n,C), it is a quotient (cf. [291]) of the Stiefel manifold of frames St(n,C∞):= {(v1,...,v n)|vi∈C∞,rankv1,...,v n=n}. Clearly the universal vector subbundle on the Grassmannian is obtained as the invariant direct image of a GL(n,C)- linearised vector bundle on the Stiefel manifold. The names change a little bit: instead of a linearized vector bundle one talks here of G-equivariant vector bundles, and there is a theory of equivariant characteristic classes, i.e., these classes are equivariant for morphisms of G-spaces (spaces with a G-action). Rather than dwelling more on abstract definitions and general properties, I prefer at this stage to mention the most important issues of G-linearized bundles for geometric rational K(π, 1)’s. 6.10 Hodge bundles of weight = 1 (1) Hodge bundle for families of Abelian varieties Consider the family of Abelian Varieties of dimension g, and with a polarization of type D∼(d1,d2,...,dg)over Siegel’s upper half space Hg:= τ∈Mat(g,g,C)|τ=tτ,Im(τ ) > 0. The family is given by the quotient of the trivial bundle H:= Hg×Cgby the action of λ∈Z2gacting by (τ, z)→ (τ, z+(D,−τ)λ). The local system corresponding to the first cohomology groups of the fibres is the Sp(D,Z)-linearized local system (bundle with fibre Z2g)onHZ:= Hg×Z2g, with the obvious action of Sp(D,Z) M∈Sp(D,Z), M(τ, λ) =(M(τ ), Mλ), M:= αβ γδ ,M(τ ) =−D(Dα−τγ) −1(Dβ−τδ). Obviously HZ⊗ZCsplits as H1,0⊕H0,1and the above vector bundle Hequals H1,0and is called the Hodge bundle. Stacks are nowadays the language in order to be able to treat the above local system and the Hodge bundle as living over the quotient moduli space Ag,D:= Hg/Sp(D,Z). 123
346 F. Catanese (2) Hodge Bundle for families of curves Over Teichmüller space we have a universal family of curves, which we shall denote by pg:Cg→Tg. The reason is the following: letting Mbe here a compact oriented Riemann surface of genus g, and C(M)the space of complex structures on M, it is clear that there is a universal tautological family of complex structures parametrized by C(M), and with total space UC(M):= M×C(M), on which the group Diff+(M)naturally acts, in particular Diff0(M). A rather simple consequence of the Lefschetz’ Lemma 34 is that Diff0(M)acts freely on C(M): in fact Lefschetz’ Lemma 34 implies that for each complex structure Con Mthe group of biholomorphisms Aut (C)contains no automorphism (other than the identity) which is homotopic to the identity, hence a fortiori no one that is differentiably isotopic to the identity. We have the local system HZ:= R1(pg)∗(Z), and again HZ⊗ZC=H1,0⊕H0,1,H1,0:= (pg)∗1 Cg|Tg. H:= H1,0is called the Hodge bundle, and it is indeed the pull back of the Hodge bundle on Agby the Torelli map, or period map, which associates to a complex structure Cthe Jacobian variety Jac(C):= H1,0(C)∨/H1(M,Z). The i-th Chern class of the Hodge bundle yield some cohomology classes which Mumford [304,307] denoted as the λi-class (while the notation λ-class is reserved for λ1). Other classes, which play a crucial role in Mumford’s conjecture ([307], see also theorem 233), are the classes Ki, defined as Ki:= (pg)∗(Ki+1), K:= c11 Cg|Tg. 6.11 A surface in a Bagnera–De Franchis threefold We want to describe here a construction given in joint work with Ingrid Bauer and Frapporti [37]ofasurfaceSwith an ample canonical divisor KS, and with K2 S=6, pg:= h0(OS(KS)) =1,q:= h0(1 S)=1. Let A1be an elliptic curve, and let A2be an Abelian surface with a line bundle L2 yielding a polarization of type (1,2)(i.e., the elementary divisors for the Chern class of L2are d1=1,d2=2). Take as L1the line bundle OA1(2O), and let Lbe the line bundle on A:= A1×A2obtained as the exterior tensor product of L1and L2,so that H0(A,L)=H0(A1,L1)⊗H0(A2,L2). 123
Topological methods in moduli theory 347 Moreover, we choose the origin in A2so that the space of sections H0(A2,L2) consists only of even sections (hence, we shall no longer be free to further change the origin by an arbitrary translation). We want to take a Bagnera–De Franchis threefold X:= A/G, where A=(A1× A2)/T, and G∼ =T∼ =Z/2, and have a surface S⊂Xwhich is the quotient of a G×Tinvariant D∈|L|, so that S2=1 4D2=6. We write as usual A1=C/Z⊕Zτ, and we let A2=C2/2, with λ1,λ 2,λ 3,λ 4a basis of 2on which the Chern class of L2is in Frobenius normal form. We let then G:= {Id,g},g(a1+a2):= a1+τ/2−a2+λ2/2,∀a1∈A1,a2∈A2 (66) T:= (Z/2)(1/2+λ4/2)⊂A=(A1×A2). (67) Now, G×Tsurjects onto the group of two torsion points A1[2]of the elliptic curve (by the homomorphism associating to an affine transformation its translation vector) and also on the subgroup (Z/2)(λ2/2)⊕(Z/2)(λ4/2)⊂A1[2], and both H0(A1,L1) and H0(A2,L2)are the Stone–von Neumann representation of the Heisenberg group which is a central Z/2 extension of G×T. By Proposition 60, since in this case (recall the notation A=V/), we have V∼ =¯ V (the only roots of unity occurring are just ±1), we conclude that there are exactly 4 divisors in |L|, invariant by a1+a2→ a1−a2, and by a1+a2→ a1+τ/2+a2+λ2/2, and a1+a2→ a1+1/2+a2+λ4/2. Hence these four divisors descend to give four surfaces S⊂X. This construction is used in [37] to prove the following. Theorem 68 Let S be a surface of general type with invariants K 2 S=6,p g=q=1 such that there exists an unramified double cover ˆ S→S with q (ˆ S)=3, and such that the Albanese morphism ˆα:ˆ S→A is birational onto it image Z , a divisor in A with Z3=12. Then the canonical model of ˆ S is isomorphic to Z , and the canonical model of S is isomorphic to Y =Z/(Z/2), which a divisor in a Bagnera–De Franchis threefold X:= A/G, where A =(A1×A2)/T, G ∼ =T∼ =Z/2, and where the action is as in (66,67). These surfaces exist, have an irreducible four dimensional moduli space, and their Albanese map α:S→A1=A1/A1[2]has general fibre a non hyperelliptic curve of genus g =3. Proof By assumption the Albanese map ˆα:ˆ S→Ais birational onto Z, and we have K2 ˆ S=12 =K2 Z, since by adjunction OZ(Z)is the dualizing sheaf of Z(so Zrestricts to the canonical divisor KZof Z). We argue similarly to [35], Step 4 of theorem 0.5, p. 31. Denote by Wthe canonical model of ˆ S, and observe that by adjunction (see loc. cit.) we have KW=ˆα∗(KZ)−U, where Uis an effective Q-Cartier divisor. 123
348 F. Catanese We observe now that KZis ample (by adjunction, since Zis ample) and KWis also ample, hence we have an inequality, 12 =K2 W=(ˆα∗(KZ)−U)2=K2 Z−(ˆα∗(KZ)·U)−(KW·U)≥K2 Z=12, and since both terms are equal to 12, we conclude that U=0, which means that KZpulls back to KWhence Wis isomorphic to Z. We have a covering involution ι:ˆ S→ˆ S, such that S=ˆ S/ι. Since the action of Z/2isfreeon ˆ S,Z/2 also acts freely on Z. Since Z3=12, Zis a divisor of type (1,1,2)in A. The covering involution ι: ˆ S→ˆ Scan be lifted to an involution gof A, which we write as an affine transformation g(a)=αa+β. We have Abelian subvarieties A1=ker(α −Id),A2=ker(α +Id), and since the irregularity of Sequals 1, A1has dimension 1, and A2has dimension 2. We observe preliminarly that gis fixed point free: since otherwise the fixed point locus would be non empty of dimension one, so it would intersect the ample divisor Z, contradicting that ι:Z→Zacts freely. Therefore Y=Z/ι is a divisor in the Bagnera–De Franchis threefold X=A/G, where Gis the group of order two generated by g. We can then write (using the notation introduced in Proposition 16) the Abelian threefold Aas A/T, and since β1/∈T1we have only two possible cases. Case (0) : T=0. Case (1) : T∼ =Z/2. We further observe that since the divisor Zis g-invariant, its polarization is αinvariant, in particular its Chern class c∈∧ 2(Hom(, Z)), where A=V/. Since T=/(1⊕2),cpulls back to c∈∧ 2(Hom(1⊕2,Z)) =∧ 2(∨ 1)⊕∧ 2(∨ 2)⊕(∨ 1)⊗(∨ 2), and by invariance c=(c 1⊕c 2)∈∧ 2(∨ 1)⊕∧ 2(∨ 2). So Case 0) bifurcates in the cases: Case (0-I) c 1is of type (1),c 2is of type (1,2). Case (0-II) c 1is of type (2),c 2is of type (1,1). Both cases can be discarded, since they lead to the same contradiction. Set D:= Z: then Dis the divisor of zeros on A=A1×A2of a section of a line bundle Lwhich is an exterior tensor product of L1and L2. Since H0(A,L)=H0(A1,L1)⊗H0(A2,L2), and H0(A1,L1)has dimension one in case (0-I), while H0(A2,L2)has dimension one in case (0-II), we conclude that Dis a reducible divisor, a contradiction, since D is smooth and connected. In case (1), we denote A:= A1×A2, and we let Dbe the inverse image of Zinside A.AgainDis smooth and connected, since π1(ˆ S)surjects onto .Now D2=24, so the Pfaffian of cequals 4, and there are a priori several possibilities. 123
Topological methods in moduli theory 349 Case (1-I) c 1is of type (1). Case (1-II) c 2is of type (1,1). Case (1-III) c 1is of type (2),c 2is of type (1,2). Cases (1-I) and (1-II) can be excluded as case (0), since then Dwould be reducible. We are then left only with case (1-III), and we may, without loss of generality, assume that H0(A1,L1)=H0(A1,OA1(2O)), and assume that we have chosen the origin so that all the sections of H0(A2,L2)are even. We have A=A/T, and we may write the generator of Tas t1⊕t2, and write g(a1+a2)=(a1+β1)+(a2−β2). By the description of Bagnera–De Franchis varieties given in Sect. 5.1 we have that t1and β1are a basis of the group of 2 torsion points of the elliptic curve A1. Now our divisor D, since all sections of L2are even, is G×Tinvariant if and only if it is invariant by Tand by translation by β. This condition however implies that translation by β2of L2is isomorphic to L2, and similarly for t2.Itfollowsthatβ2,t2are basis of the kernel K2of the map φL2: A2→Pic0(A2), associating to ythe tensor product of the translation of L2by ywith the inverse of L1. The isomorphism of G×Twith both K1:= A1[2]and K2allows to identify both H0(A1,L1)and H0(A2,L2)with the Stone von Neumann representation L2(T): observe in fact that there is only one alternating function (G×T)→Z/2, independent of the chosen basis. Therefore, there are exactly 4 invariant divisors in the linear system |L|. Explicitly, if H0(A1,L1)has basis x0,x1and H0(A2,L2)has basis y0,y1, then the invariant divisors correspond to the four eigenvectors x0y0+x1y1,x0y0−x1y1,x0y1+x1y0,x0y1−x1y0. To prove irreducibility of the above family of surfaces, it suffices to show that all the four invariant divisors occur in the same connected family. To this purpose, we just observe that the monodromy of the family of elliptic curves Eτ:= C/(Z⊕Zτ) on the upperhalf plane has the effect that a transformation in SL(2,Z)acts on the subgroup Eτ[2]of points of 2-torsion by its image matrix in GL(2,Z/2), and in turn the effect on the Stone von Neumann representation is the one of twisting it by a character of Eτ[2]. This concludes the proof that the moduli space is irreducible of dimension 4, since the moduli space of elliptic curves, respectively the moduli space of Abelian surfaces with a polarization of type (1,2), are irreducible, of respective dimensions 1,3. The final assertion is a consequence of the fact that Alb(S)=A1/(T1+β1), so that the fibres of the Albanese map are just divisors in A2of type (1,2). Their self intersection equals 4 =2(g−1), hence g=3. In order to establish that the general fibre of the Albanese map is non hyperelliptic, it suffices to prove the following lemma. Lemma 69 Let A2be an Abelian surface, endowed with a divisor L of type (1,2), so that there is an isogeny of degree two f :A2→Aonto a principally polarised Abelian surface, with kernel generated by a point t of 2-torsion, and such that L = 123
350 F. Catanese f∗(). Then the only curves C ∈|L|that are hyperelliptic are contained in the pull backs of a translate of by a point of order 2for a suitable such isogeny f :A2→A. In particular, the general curve C ∈|L|is not hyperelliptic. Proof Let C∈|L|, and consider D:= f∗(C)∈|2|. There are two cases. Case (I): C+t=C. Then D=2B, where Bhas genus 2, so that C=f∗(B), hence, since 2B≡2,Bis a translate of by a point of order 2. There are exactly two such curves, and for them C→Bis étale. Case (II): C+t= C. Then C→Dis birational, f∗(D)=C∪(C+t).Now, C+tis also linearly equivalent to L, hence C∩(C+t)meet in the 4 base points of the pencil |L|. Hence Dhas two double points and geometric genus equal to 3. These double points are the intersection points of and a translate by a point of order 2, and are points of 2-torsion. The sections of H0(OA(2)) are all even and |2|is the pull-back of the space of hyperplane sections of the Kummer surface K⊂P3, the quotient K=A/{±1}. Therefore the image Eof each such curve Dlies in the pencil of planes through 2 nodes of K. Eis a plane quartic, hence Ehas geometric genus 1, and we conclude that C admits an involution σwith quotient an elliptic curve E(normalization of E), and the double cover is branched in 4 points. Assume that Cis hyperelliptic, and denote by hthe hyperelliptic involution, which lies in the centre of Aut(C). Hence we have (Z/2)2acting on C, with quotient P1. We easily see that there are exactly six branch points, two being the branch points of C/h→P1, four being the branch points of E→P1. It follows that there is an étale quotient C→B, where Bis the genus 2 curve, double cover of P1branched on the six points. Now, the inclusion C⊂A2and the degree 2 map C→Binduces a degree two isogeny A2→J(B), and Cis the pull back of the Theta divisor of J(B), thus it cannot be a general curve. QED for the lemma. Definition 70 Let us call a surface Sas in Theorem 68 aSicilian surface with q= pg=1. Observe that the fundamental group of Sis isomorphic to the fundamental group of X, and that , fitting into the exact sequence 1→→→G=Z/2→1, is generated by the union of the set {g,t},where g(v1+v2)=v1+τ/2−v2+λ2/2 t(v1+v2)=v1+1/2+v2+λ4/2 with the set of translations by the elements of a basis λ1,λ 2,λ 3,λ 4of 2. It is therefore a semidirect product of Z5=2⊕Ztwith the infinite cyclic group generated by g: conjugation by gacts as −1on2, and it sends t→ t−λ4(hence 2t−λ4is an eigenvector for the eigenvalue 1). 123
Topological methods in moduli theory 351 We shall now give a topological characterization of Sicilian surfaces with q= pg=1, following the lines of [222]. Observe in this respect that Xis a K(, 1)space, so that its cohomology and homology are just group cohomology, respectively homology, of the group . Corollary 71 A Sicilian surface S with q =pg=1is characterized by the following properties: (1) K2 S=6 (2) χ(S)=1 (3) π1(S)∼ =, where is as above, (4) the classifying map f :S→X , where X is the Bagnera–De Franchis threefold which is a classifying space for , has the property that f∗[S]:=Y satisfies Y3=6. In particular, any surface homotopically equivalent to a Sicilian surface is a Sicilian surface, and we get a connected component of the moduli space of surfaces of general type which is stable under the action of the absolute Galois group.7 Proof Since π1(S)∼ =,firstofallq(S)=1, hence also pg(S)=1. By the same token there is a double étale cover ˆ S→Ssuch that q(ˆ S)=3, and the Albanese image of ˆ S, counted with multiplicity, is the inverse image Zof Y, therefore Z3=12. From this, it follows that ˆ S→Zis birational, since the class of Zis indivisible. We may now apply the previous theorem in order to obtain the classification. Observe finally that the condition (ˆα∗ˆ S)3=12 is not only a topological condition, it is also invariant under Galois autorphisms. 7 Regularity of classifying maps and fundamental groups of projective varieties 7.1 Harmonic maps Given a continuous map f:M→Nof differentiable manifolds, we can approximate it, as already partly explained, by a differentiable one, homotopic to the previous one. Indeed, as we already explained, we may assume that N⊂Rn,M⊂Rmand, by a partition of unity argument, that Mis an open set in Rh. Convolution approximates then f by a differentiable function F1with values in a tubular neighbourhood T(N) of N, and then the implicit function theorem applied to the normal bundle provides a differentiable retraction r:T(N)→N. Then F:= r◦F1is the required approxima- tion, and the same retraction provides a homotopy between fand F(the homotopy between fand F1being obvious). If however M,Nare algebraic varieties, and algebraic topology tells us about the existence of a continuous map fas above, we would wish for more regularity, possibly holomorphicity of the homotopic map F. 7As we shall see in the last sections of the article, the absolute Galois group permutes the connected components of the moduli space of surfaces of general type, and this action is indeed faithful. 123
352 F. Catanese Now, Wirtinger’s theorem (see [306]) characterises complex submanifolds as area minimizing ones, so the first idea is to try to deform a differentiable mapping funtil it minimizes some functional. We may take the Riemannian structure inherited form the chosen embedding, and assume that (M,gM), (N,gN)are Riemannian manifolds. If we assume that Mis compact, then one defines the Energy E(f)of the map as the integral: E(f):= 1/2M|Df|2dμM, where Df is the derivative of the differentiable map f,dμMis the volume element on M, and |Df|is just its norm as a differentiable section of a bundle endowed with a metric: Df ∈H0(M,C∞(TM ∨⊗f∗(TN))). Remark 72 (1) in more concrete terms, the integral in local coordinates has the form E(f):= 1/2M α,β,i,j(gN)α,β ∂fα ∂xi ∂fβ ∂xj (gM)−1 i,jdet(gij). (2) Linear algebra shows that, once we identify TM,TN with their dual bundles via the Riemannian metrics, |Df|2=Tr(( Df)∨◦Df), hence we integrate over M the sum of the eigenvalues of the endomorphism (Df)∨◦Df :TM →TM. (3) The first variation of the energy function vanishes precisely when fis a harmonic map, i.e., ( f)=0, where ( f):= Tr(∇(Df)), ∇being the connection on TM ∨⊗f∗(TN)induced by the Levi-Civita connections on Mand N. (4) The energy functional enters also in the study of geodesics and Morse theory (see [288]) These notions were introduced by Eells and Sampson in the seminal paper [147], whichusedtheheat flow ∂ft ∂t=( f) in order to find extremals for the energy functional. These curves in the space of maps are (as explained in [147]) the analogue of gradient lines in Morse theory, and the energy functional decreases on these lines. The obvious advantage of the flow method with respect to discrete convergence procedures (‘direct methods of the calculus of variations’) is that here it is clear that all the maps are homotopic to each other!8 The next theorem is one of the most important results, first obtained in [147] 8The flow method made then its way further through the work of Hamilton [206], Perelman and others [320–322], leading to the solution of the three dimensional Poincaré conjecture (see for example [299]for an exposition). 123
Topological methods in moduli theory 353 Theorem 73 (Eells–Sampson) Let M,N be compact Riemannian manifolds, and assume that the sectional curvature KNof N is semi-negative (KN≤0): then every continuous map f0:M→N is homotopic to a harmonic map f :M→N. More- over the equation ( f)=0implies, in case where M,N are real analytic manifolds, the real analyticity of f . Remark 74 Observe that, in the case where Nhas strictly negative sectional curvature, Hartmann [209] proved the unicity of the harmonic map in each homotopy class. Not only the condition about the curvature is necessary for the existence of a har- monic representative in each homotopy class, but moreover it constitutes the main source of connections with the concept of classifying spaces, in view of the clas- sical (see [76,288]) theorem of Cartan–Hadamard establishing a deep link between curvature and topology. Theorem 75 (Cartan–Hadamard) Suppose that N is a complete Riemannian mani- fold, with semi-negative (KN≤0) sectional curvature: then the universal covering ˜ N is diffeomorphic to an Euclidean space, more precisely given any two points there is a unique geodesic joining them. Remark 76 The reader will notice that the hypotheses of Theorem 73 and of Hart- mann’s theorem apply naturally to two projective curves M,Nof the same genus g≥2, taken with the metric of constant curvature −1 provided by the uniformization theorem: then one may take for f0a diffeomorphism, and apply the result, obtaining a unique harmonic map fwhich Samson [330] shows to be also a diffeomorphism. The result obtained is that to the harmonic map fone associates a quadratic differential ηf∈H0(⊗2 M)=H0(OM(2K)), and that ηfdetermines the isomorphism class of N. This result constitutes another approach to Teichmüller space Tg. Thus in complex dimension 1 one cannot hope for a stronger result, to have a holomorphic map rather than just a harmonic one. The surprise comes from the fact that, with suitable assumptions, the hope can be realized in higher dimensions, with a small proviso: given a complex manifold X, one can define the conjugate manifold ¯ Xas the same differentiable manifold, but where in the decomposition TX⊗RC= T(1,0)⊕T(0,1)the roles of T(1,0)and T(0,1)are interchanged (this amounts, in case where Xis an algebraic variety, to replacing the defining polynomial equations by polynomials obtained from the previous ones by applying complex conjugation to the coefficients, i.e., replacing each P(x0,...,xN)by P(x0,...,xN)). In this case the identity map, viewed as a map ι:X→¯ Xis no longer holomorphic, but antiholomorphic. Assume now that we have a harmonic map f:Y→X: then also ι◦fshall be harmonic, but a theorem implying that fmust be holomorphic then necessarily implies that there is a complex isomorphism between Xand ¯ X. Unfortu- nately, this is not the case, as one sees, already in the case of elliptic curves; but then one may restrict the hope to proving that fis either holomorphic or antiholomorphic. A breakthrough in this direction was obtained by Siu [344] who proved several results, that we shall discuss in the next sections. 123
354 F. Catanese 7.2 Kähler manifolds and some archetypal theorem The assumption that a complex manifold Xis a Kähler manifold is that there exists a Hermitian metric on the tangent bundle T(1,0)whose associated (1,1)form ξis closed. In local coordinates the metric is given by h=i,jgi,jdzid¯zj,with dξ=0,ξ:= (i,jgi,jdzi∧d¯zj). Hodge theory shows that the cohomology of a compact Kähler manifold Xhas a Hodge–Kähler decomposition, where Hp,qis the space of harmonic forms of type (p,q), which are in particular d-closed (and d∗-closed): Hm(X,C)=⊕p,q≥0,p+q=mHp,q,Hq,p=Hp,q,Hp,q∼ =Hq(X,p X). We give just an elementary application of the above theorem, a characterization of complex tori (see [90,103,110] for other characterizations) Theorem 77 Let X be a cKM, i.e., a compact Kähler manifold X, of dimension n. Then X is a complex torus if and only if it has the same integral cohomology algebra of a complex torus, i.e. H ∗(X,Z)∼ =∧∗H1(X,Z). Equivalently, if and only if H∗(X,C)∼ =∧∗H1(X,C)and H2n(X,Z)∼ =∧2nH1(X,Z) Proof Since H2n(X,Z)∼ =Z, it follows that H1(X,Z)is free of rank equal to 2n, therefore dimC(H1,0)=n. We consider then, chosen a base point x0∈X,the Albanese map aX:X→Alb(X):= H0(1 X)∨/Hom(H1(X,Z), Z), x→ x x0 . Therefore we have a map between Xand the complex torus T:= Alb(X), which induces an isomorphism of first cohomology groups, and has degree 1, in view of the isomorphism H2n(X,Z)∼ =2n(H1(X,Z)) ∼ =H2n(T,Z). In view of the normality of X, it suffices to show that aXis finite. Let Ybe a subvariety of Xof dimension m>0 mapping to a point: then the cohomology (or homology class, in view of Poincaré duality) class of Yis trivial, since the cohomology algebra of Xand Tare isomorphic. But since Xis Kähler, if ξis the Kähler form, Yξm>0, a contradiction, since this integral depends only (by the closedness of ξ) on the homology class of Y. One can conjecture that a stronger theorem holds, namely Conjecture 78 Let X be a cKM, i.e., a compact Kähler manifold X , of dimension n. Then X is a complex torus if and only if it has the same rational cohomology algebra of a complex torus, i.e. H ∗(X,Q)∼ =∧∗H1(X,Q). Equivalently, if and only if H∗(X,C)∼ =∧∗H1(X,C). 123
Topological methods in moduli theory 355 Remark 79 Observe that H∗(X,Q)∼ =∧∗H1(X,Q)⇔H∗(X,C)∼ =∧∗H1(X,C) by virtue of the universal coefficients theorem. The same argument of the previous Theorem 77 yields that, since H2n(X,Z)∼ =Z, dimC(H1,0)=nand the Albanese map aX:X→A:= Alb(X):= H0(1 X)∨/Hom(H1(X,Z), Z), x→ x x0 is finite, and it suffices to show that it is unramified (étale), since we have an isomor- phism H1(X,Z)∼ =H1(A,Z). One sets therefore Rto be the ramification divisor of aX, and B=aX∗(R)the branch divisor. There are two cases: Case (i) Bis an ample divisor, and Aand Xare projective. Case (ii) : Bis non ample, thus, by a result of Ueno [358] there is a subtorus A0⊂A such that Bis the pull-back of an ample divisor on A/A0. Case (ii) can be reduced, via Ueno’s fibration to case (i), which is the crucial one. Since we have an isomorphism of rational cohomology groups, and Poincaré dual- ity holds, we get that aX∗and a∗ Xare, in each degree, isomorphisms with rational coefficients. The first question is to show that the canonical divisor, which is just the ramification divisor R, is an ample divisor. Then to observe that the vanishing of the topological Euler Poincaré characteristic e(X)and of the Euler Poincaré characteristics χ(i X)=0 for all iimplies (using Riemann Roch?) that the first Chern class of Xis zero, thus obtaining a contradiction to the ampleness of KX. This step works at least in dimension n=2, since then we have the Noether formula e(X)+K2 X=12χ(OX), hence we obtain K2 X=0. Added in proof: Will Sawin observed that case i) does not occur for varieties of general type, since for these, by a result of Popa and Schnell, holomorphic 1- forms always have zeros. Instead, Debarre, Jiang and Lahoz showed that case ii) leads to the occurrence of certain iterated torus bundles, so that only a proper modification of our conjecture can be true. Remark 80 Of course the hypothesis that Xis Kähler is crucial: there are several examples, due to Blanchard, Calabi, and Sommese [46–48,69,348], of complex man- ifolds which are diffeomorphic to a complex torus but are not complex tori: indeed KX is not linearly equivalent to a trivial divisor (see [99] for references to the cited papers , and [110] for partial results on the question whether a compact complex manifold with trivial canonical divisor, which is diffeomorphic to a torus, is indeed biholomorphic to a complex torus). The previous Theorem 77 allows a simple generalization which illustrates well the use of topological methods in moduli theory. 123
356 F. Catanese Theorem 81 Let X =A/G be a Generalized Hyperelliptic manifold of complex dimension n and assume that Y is a compact Kähler Manifold satisfying the following properties: (1) π1(Y)∼ =π1(X)∼ = (2) H2n(Y,Z)∼ =H2n(, Z)via the natural homomorphism of cohomology groups induced by the continuous map of Y to the classifying space X =B, associated to the homomorphism π1(Y)∼ =π1(X)∼ =. (3) Hi(Y,C)∼ =Hi(, C)∀i via the natural homomorphism analogous to the one defined in (2). Then (1) Y is a Generalized Hyperelliptic manifold Y =A/G, (2) if G is abelian, then Y is (real)affinely equivalent to X , and moreover (3) Y is a complex deformation of X if and only if X and Y have the same Hodge type. Proof Without loss of generality, we may assume that Gcontains no translations. Since =π1(X), we have an exact sequence 1 →→→G→0, and we let Wbe the unramified covering of Ycorresponding to the surjection π1(Y)∼ =→G. Then the Albanese variety Aof Whas dimension n, and the group Gacts on A. Main claim: Gacts freely on A. This follows since, otherwise, there is an element g∈G\1Gand a lift gof gin such that a positive power gmof gequals the neutral element 1(see Step I of the proof of Proposition 21). But then gwould not act freely on A. Now the Albanese map W→Ainduces a holomorphic map f:Y→A/G, and we know that A/Gis a classifying space with π1(A/G)∼ =. By our hypothesis f induces an isomorphism of cohomology groups and has degree equal to 1. The same argument as in Theorem 77 shows that fis an isomorphism. Therefore Yis also a Generalized Hyperelliptic manifold of complex dimension n. For the second assertion, we simply apply Proposition 21, stating that the exact sequence 1 →→→G→0 determines, when Gis abelian, the real affine type of the action of . The last assertion is a direct consequence of Remark 20. The following is one more characterization of complex tori and Abelian varieties. Corollary 82 Let Z be a projective variety. Then Z is a projective K (π , 1)with π an abelian group ⇔Z is an Abelian variety. Similarly, if Z is a cKM, Z is a K (π, 1) with πabelian ⇔Z is a complex torus. Proof Since πis finitely generated, we can write π=Zm⊕T, where Tis a finite abelian group. The subgroup Zmis the fundamental group of a Galois cover W→Z with group T. In this case mmust be even m=2n, by the Kähler assumption, and since H∗(W,Z)=H∗(Z2n,Z), we obtain that n=dimC(W)=dimC(Z)and we can apply Theorem 77 to infer that Wis a complex torus. Now, Z=W/T, where Tis abelian, and Tacts trivially by conjugation on π1(W)∼ =Z2n. Hence Tis a group of translations, and we obtain that Zis a complex torus. Moreover, since the fundamental group of a torus is torsion free, we can actually conclude that T=0. 123
Topological methods in moduli theory 357 7.3 Siu’s results on harmonic maps The result by Siu that is the simplest to state is the following Theorem 83 (I) Assume that f :M→N is a harmonic map between two compact Kähler manifolds and that the curvature tensor of N is strongly negative. Assume further that the real rank of the derivative D f is at least 4in some point of M. Then f is either holomorphic or antiholomorphic. (II) In particular, if dimC(N)≥2and M is homotopy equivalent to N, then M is either biholomorphic or antibiholomorphic to N . From Theorem 83 follows an important consequence. Corollary 84 Assume that f :M→N is a continuous map between two compact Kähler manifolds and that the curvature tensor of N is strongly negative. Assume further that there is a j ≥4such that H j(f,Z)= 0: then f is homotopic to a map F which is either holomorphic or antiholomorphic. Proof f is homotopic to a harmonic map F. One needs to show that at some point the real rank of the derivative DF is at least 4. If it were not so, then by Sard’s lemma the image Y:= F(M)would be a subanalytic compact set of Hausdorff dimension at most 3. Lemma 85 Let Y be a subanalytic compact set, or just a compact set of Hausdorff dimension at most h. Then Hi(Y,Z)=0for i ≥h+1. The lemma then implies Hi(Y,Z)=0,∀i≥4, contradicting the existence of an i≥4 such that Hi(f,Z)= 0. First proof of the lemma: Y is a finite union of locally closed submanifolds of dimension ≤h, by the results of [42]. Using the exact sequence of Borel-Moore homology (see [59]) relating the homology of Y=U∪F, where Yis the union of aclosedsetFwith an open subset U, and induction on the number of such locally closed sets, we obtain that Hi(Y,Z)=0fori≥h+1. Second proof of the lemma: Y is a compact of Hausdorff dimension at most h.By theorem VII 3 of [219], page 104, follows that the dimension of Yis at most h.In turn, using theorem 2’ , p. 362 of [1], it follows (take Aibidem to be a point P) that Hi(Y,Z)=Hi(Y,P,Z)=0fori≥h+1. Remark 86 (1) The hypothesis that the rank of the differential should be at least 4 at some point is needed for instance to avoid that the image of Fis a complex projective curve Ccontained inside a projective variety Nwith such a strongly negative Kähler metric. Because in this case fcould be the composition of a holomorphic map g:M→Cto a curve Cof the same genus as C, but not isomorphic to C, composed with a diffeomorphism between Cand C. (2) Part II of Theorem 83 follows clearly from part I: because, as we have seen, the homotopy equivalence between Mand Nis realised by a harmonic map f:M→N. From part I of the theorem one knows that fis holomorphic or antiholomorphic (in short, one says that fis dianalytic). W.l.o.g. let us assume that fis holomorphic 123
358 F. Catanese (replacing possibly Nby ¯ N). There remains only to prove that fis biholomorphic. The argument is almost the same as the one used in the archetypal Theorem 77. Step 1: fis a finite map; otherwise, since fis proper, there would be a complex curve Csuch that f(C)is a point. But in a Kähler manifold the homology class in dimension 2mof a subvariety of complex dimension m, here C, is never trivial. On the other hand, since fis a homotopy equivalence, and f(C)is a point, this class must be zero. Step 2: fis a map of degree one since finduces an isomorphism between the last non zero homology groups of M,Nrespectively. If n=dimCN,m=dimCM, then these groups are H2m(M,Z), respectively H2n(N,Z); hence n=mand H2m(f):H2m(M,Z)∼ =Z[M]→H2m(N,Z)∼ =Z[N] is an isomorphism. Step 3: fis holomorphic, finite and of degree 1.Therefore there are open subsets whose complements are Zariski closed such that f:U→V⊂Nis an isomorphism. Then the inverse of fis defined on V, the complement of a complex analytic set, and by the Riemann extension theorem (normality of smooth varieties) the inverse extends to N, showing that fis biholomorphic. (3) We have sketched the above argument since it appears over and over in the application of homotopy equivalence to proving isomorphism of complex projective (or just Kähler) manifolds. Instead, the proof of part I is based on the Bochner–Nakano formula, which was later further generalised by Siu [346], and is too technical to be fully discussed here. Let us try however to describe precisely the main hypothesis of strong negativity of the curvature, which is a stronger condition than the strict negativity of the sectional curvature. As we already mentioned, the assumption that Nis a Kähler manifold is that there exists a Hermitian metric on the tangent bundle T(1,0)whose associated (1,1)form is closed. In local coordinates the metric is given by i,jgi,jdzid¯zj,with d(i,jgi,jdzi∧d¯zj)=0. The curvature tensor is a (1,1)form with values in (T(1,0))∨⊗T(1,0), and using the Hermitian metric to identify (T(1,0))∨∼ =T(1,0)=T(0,1), and their conjugates ((T(0,1))∨=(T(0,1))∼ =T(1,0)) we write as usual the curvature tensor as a section R of (T(1,0))∨⊗(T(0,1))∨⊗(T(1,0))∨⊗(T(0,1))∨. Then seminegativity of the sectional curvature is equivalent to −Rξ∧¯η−η∧¯ ξ,ξ∧¯η−η∧¯ ξ≤0, 123
Topological methods in moduli theory 359 for all pairs of complex tangent vectors ξ,η (here one uses the isomorphism T(1,0)∼ = TN, and one sees that the expression depends only on the real span of the two vectors ξ,η). Strong negativity means instead that −Rξ∧¯η−ζ∧¯ θ,ξ∧¯η−ζ∧¯ θ<0, for all 4-tuples of complex tangent vectors ξ,η,ζ, θ. The geometrical meaning is the following (see [2], p. 71): the sectional curvature is a quadratic form on ∧2(TN), and as such it extends to the complexified bundle ∧2(TN)⊗Cas a Hermitian form. Then strong negativity in the sense of Siu is also called negativity of the Hermitian sectional curvature R(v, w, ¯v, ¯w) for all vectors v, w ∈(TN)⊗C. Then a reformulation of the result of Siu [344] and Sampson [331] is the following: Theorem 87 Let M be a compact Kähler manifold, and N a Riemannian manifold with semi-negative Hermitian sectional curvature. Then every harmonic map f : M→N is pluri-harmonic. Now, examples of varieties Nwith a strongly negative curvature are the balls in Cn, i.e., the BSD of type In,1; Siu finds out that ([344],seealso[70]) for the irreducible bounded symmetric domains of type Ip,q,for pq ≥2,II n,∀n≥3,III n,∀n≥2,IV n,∀n≥3, the metric is not strongly negative, but just very strongly seminegative, where very strong negativity simply means negativity of the curvature as a Hermitian form on T1,0⊗T0,1=T1,0⊗T0,1. Indeed, the bulk of the calculations is to see that there is an upper bound for the nullity of the Hermitian sectional curvature, i.e. for the rank of the real subbundles of TM where the Hermitian sectional curvature restricts identically to zero (in Siu’s notation, then one considers always the case where very strongly seminegativity holds, and 2-negative means strongly negative, adequately negative means that the nullity cannot be maximal). Hence Siu derives several results, Theorem 88 (Siu) Suppose that M ,N are compact Kähler manifolds and the cur- vature tensor of N is negative of order k. Assume that f :M→N is a harmonic map such that, ∃i≥2k such that Hi(f,Z)= 0. Then f is either holomorphic or antiholomorphic. Theorem 89 (Siu) Suppose that M is a compact Kähler manifold and N is a locally symmetric manifold D/, where Dis an irreducible bounded symmetric domain of one of the following types: Ip,q,for pq ≥2,II n,∀n≥3,III n,∀n≥2,IV n,∀n≥3. 123
360 F. Catanese Assume that f :M→N is a harmonic map such that, n := dimCN, H 2n(f,Z)= 0. Then f is either holomorphic or antiholomorphic. In particular, if M is a compact Kähler manifold homotopically equivalent to N as above, then either M ∼ =NorM∼ =¯ N. The most general result is ([2], p. 80, theorems 6.13–15): Theorem 90 (Siu) Suppose that M is a compact Kähler manifold and N is a locally Hermitian symmetric space D/ =G/K , where the irreducible decomposition of D contains no dimension 1 factors. (1) Assume that f :M→N is a harmonic map such that, n := dimCN, H2n(f,Z)= 0. Then f is holomorphic for some invariant complex structure on G/K. (2) Assume that M is homotopically equivalent to N : then f is biholomorphic to G/K for some invariant complex structure on G/K. Remark 91 A few words are needed to explain the formulation ‘for some invariant complex structure on G/K’. In the case where Dis irreducible, we can just take the conjugate complex structure. But if D=D1×···×Dl, one can just take the conjugate complex structure on a subset Iof the indices j∈{1,...,l}. 7.4 Hodge theory and existence of maps to curves Siu also used harmonic theory in order to construct holomorphic maps from Kähler manifolds to projective curves. The first result in this direction was the theorem of [347], also obtained by Jost and Yau (see [239] and also [228] for other results). Theorem 92 (Siu) Assume that a compact Kähler manifold X is such that there is a surjection φ:π1(X)→πg, where g ≥2and, as usual, πgis the fundamental group of a projective curve of genus g. Then there is a projective curve C of genus g≥g and a fibration f :X→C (i.e., the fibres of f are connected) such that φfactors through π1(f). In this case the homomorphism leads to a harmonic map to a curve, and one has to show that the Stein factorization yields a map to some Riemann surface which is holomorphic for some complex structure on the target. In this case it can be seen more directly how the Kähler assumption, which boils down to Kähler identities, is used. Recall that Hodge theory shows that the cohomology of a compact Kähler manifold Xhas a Hodge–Kähler decomposition, where Hp,qis the space of harmonic forms of type (p,q): Hm(X,C)=⊕p,q≥0,p+q=mHp,q,Hq,p=Hp,q,Hp,q∼ =Hq(X,p X). The Hodge–Kähler decomposition theorem has a long story, revived by Griffths in [192,193], and was proven by Picard in special cases. It entails the following conse- quence: 123
Topological methods in moduli theory 361 Holomorphic forms are closed, i.e., η∈H0(X,p X)⇒dη=0. At the turn of last century this fact was then used by Castelnuovo and de Franchis [80, 130]: Theorem 93 (Castelnuovo–de Franchis) Assume that X is a compact Kähler mani- fold, η1,η 2∈H0(X, 1 X)are C-linearly independent, and the wedge product η1∧η2 is d-exact. Then η1∧η2≡0and there exists a fibration f :X→C such that η1,η 2∈f∗H0(C, 1 C). In particular, C has genus g ≥2. Even if the proof is well known, let us point out that the first assertion follows from the Hodge–Kähler decomposition, while η1∧η2≡0 implies the existence of a non constant rational function ϕsuch that η2=ϕη1. This shows that the foliation defined by the two holomorphic forms has Zariski closed leaves, and the rest follows then rather directly taking the Stein factorization of the rational map ϕ:X→P1. Now, the above result, which is holomorphic in nature, combined with the Hodge decomposition, produces results which are topological in nature (they actually only depend on the cohomology algebra structure of H∗(X,C)). To explain this in the most elementary case, we start from the following simple observation. If two linear independent vectors in the first cohomology group H1(X,C) of a Kähler manifold have wedge product which is trivial in cohomology, and we represent them as η1+ω1,η 2+ω2,for η1,η 2,ω 1,ω 2∈H0(X, 1 X), then by the Hodge decomposition and the first assertion of the theorem of Castelnuovo–de Franchis (η1+ω1)∧(η2+ω2)=0∈H2(X,C) implies η1∧η2≡0,ω 1∧ω2≡0. We can apply Castelnuovo–de Franchis unless η1,η 2are C-linearly dependent, and similarly ω1,ω 2. W.l.o.g. we may assume η2≡0 and ω1≡0. But then η1∧ω2=0 implies that the Hodge norm X (η1∧ω2)∧(η1∧ω2)∧ξn−2=0, where ξis here the Kähler form. A simple trick is to observe that 0=X (η1∧ω2)∧(η1∧ω2)∧ξn−2=−X (η1∧ω2)∧(η1∧ω2)∧ξn−2, therefore the same integral yields that the Hodge norm of η1∧ω2is zero, hence η1∧ω2≡0;the final conclusion is that we can in any case apply Castelnuovo-de Franchis and find a map to a projective curve Cof genus g≥2. More precisely, one gets the following theorem [87]: 123
362 F. Catanese Theorem 94 (Isotropic subspace theorem) On a compact Kähler manifold X there is a bijection between isomorphism classes of fibrations f :X→C to a projective curve of genus g ≥2, and real subspaces V ⊂H1(X,C)(‘real’ means that V is self conjugate, V=V)which have dimension 2g and are of the form V =U⊕¯ U , where U is a maximal isotropic subspace for the wedge product H1(X,C)×H1(X,C)→H2(X,C). It is interesting that the above result implies the following theorem of Gromov [196], which in turn obviously implies theorem 92 of Siu (see [25,197] for related results). Theorem 95 (Gromov’s few relations theorem) Let X be a compact Kähler manifold and assume that there exists a surjection of its fundamental group := π1(X)→G=x1,...,xn|R1(x),...,Rm(x), onto a finitely presented group that has ‘few relations’, more precisely where n ≥ m−2. Then there exists a fibration f :X→C , onto a projective curve C of genus g≥1 2(n−m). If moreover π1(X)∼ =G, then the first cohomology group H 1(X,C)equals f∗H1(C,C). Proof We saw in a Sect. 6.6 that if the fundamental group of X,:= π1(X)admits a surjection onto G, then the induced classifying continuous map φ:X→BG has the properties that its induced action on first cohomology H1(φ) :H1(BG,C)→H1(X,C) is injective and the image Wof H1(φ) is such that each element w∈Wis contained in an isotropic subspace of rank ≥n−m. Hence it follows immediately that there is an isotropic subspace of dimension ≥n−mand a fibration onto a curve of genus g≥n−m. For the second assertion, let Ube a maximal isotropic subspace contained in W= H1(X,C): then there exists a fibration f:X→C, onto a projective curve Cof genus g≥2 such that f∗H1(C,C)=U⊕¯ U. We are done unless U⊕¯ U= H1(X,C). But in this case we know that W=H1(X,C)is the union of such proper subspaces U⊕¯ U. These however are of the form f∗H1(C,Z)⊗C, hence they are a countable num- ber; by Baire’s theorem their union cannot be the whole C-vector space W=H1 (X,C). Not only one sees clearly how the Kähler hypothesis is used, but indeed Kato [233] and Pontecorvo [324] showed how the results are indeed false without the Kähler assumption, using twistor spaces of algebraic surfaces which are P1-bundles over a projective curve Cof genus g≥2. A simpler example was then found by Kotschick ([2], ex. 2.16, p. 28): a primary Kodaira surface (an elliptic bundle over an elliptic curve, with b1(X)=3,b2(X)=4). 123
Topological methods in moduli theory 363 We do not mention in detail generalisations of the Castelnuovo–de Franchis theory to higher dimensional targets (see [87,190,191,342]), since these shall not be used in the sequel. We want however to mention another result ([106], see also [102] for a weaker result) which again, like the isotropic subspace theorem, determines explicitly the genus of the target curve (a result which is clearly useful for classification and moduli problems). Theorem 96 Let X be a compact Kähler manifold, and let f :X→C be a fibration onto a projective curve C, of genus g, and assume that there are exactly r fibres which are multiple with multiplicities m1,...mr≥2. Then f induces an orbifold fundamental group exact sequence π1(F)→π1(X)→π1(g;m1,...mr)→0, where F is a smooth fibre of f , and π1(g;m1,...mr) := α1,β 1,...,α g,β g,γ 1,...γ r|g 1[αj,βj]r 1γi=γm1 1=···=γmr r=1. Conversely, let X be a compact Kähler manifold and let (g,m1,...mr)be a hyperbolic type, i.e., assume that 2g−2+i(1−1 mi)>0. Then each epimorphism φ:π1(X)→π1(g;m1,...mr)with finitely generated kernel is obtained from a fibration f :X→C of type (g;m1,...mr). 7.5 Restrictions on fundamental groups of projective varieties A very interesting (and still largely unanswered) question posed by Serre [335]is: Question 97 (Serre) (1) Which are the projective groups, i.e., the groups πwhich occur as fundamental groups π=π1(X)of a complex projective manifold? (2) Which are the Kähler groups, i.e., the groups πwhich occur as fundamental groups π=π1(X)of a compact Kähler manifold? Remark 98 (i) Serre himself proved (see [337]) that the answer to the first question is positive for every finite group. In this chapter we shall only limit ourselves to mention some examples and results to give a general idea, especially about the use of harmonic maps, referring the reader to the book [2], entirely dedicated to this subject. (ii) For question (1), in view of the Lefschetz hyperplane section Theorem 3, the class of projective groups πis exactly the class of groups which occur as fundamental groups π=π1(X)of a complex projective smooth surface. (iii) if πand πare projective (resp. Kähler), the same is true for the Cartesian product (take X×X). 123
364 F. Catanese The first obvious restriction for a group to be a Kähler group (only a priori the class of Kähler groups is a larger class than the one of projective groups) is that their first Betti number b1(rank of the abelianization ab =/[, ]) is even; since if =π1(X), then ab =H1(X,Z), and H1(X,Z)⊗Chas even dimension by the Hodge–Kähler decomposition. A more general restriction is that the fundamental group of a compact differen- tiable manifold Mmust be finitely presented: since by Morse theory (see [288]) M is homotopically equivalent to a finite CW-complex (Mis obtained attaching finitely many cells of dimension ≤dimR(M)). Conversely (see [333], p. 180), Theorem 99 Any finitely presented group is the fundamental group of a compact oriented 4-manifold. Recall for this the Definition 100 (Connected sum) Given two differentiable manifolds M1,M2of the same dimension n, take respective points Pi∈Mi,i=1,2 and respective open neighbourhoods Biwhich are diffeomorphic to balls, and with smooth boundary ∂Bi∼ = Sn−1. Glueing together the two manifolds with boundary Mi\Bi,∂Biwe obtain a manifold M1!M2, which is denoted by the connected sum of M1,M2and whose diffeomorphism class is independent of the choices made in the construction. Also, if M1,M2are oriented, the same holds for M1!M2, provided the diffeomorphisms ∂Bi∼ =Sn−1are chosen to be orientation preserving. Definition 101 (Free product)If, are finitely presented groups, =x1,...,xn|R1(x),...,Rm(x), =y1,..., yh|R 1(y),...,R k(y), their free product is the finitely presented group ∗=x1,...,xn,y1,...,yh|R1(x),...,Rm(x), R 1(y),..., R k(y). Idea for the proof of theorem 99: By the van Kampen theorem, the fundamental group of the union X=X1∪X2of two open sets with connected intersection X1∩X2is the quotient of the free product π1(X1)∗π1(X2)by the relations i1(γ ) =i2(γ ), ∀γ∈ π1(X1∩X2), where ij:π1(X1∩X2)→π1(Xj)is induced by the natural inclusion (X1∩X2)⊂Xj. Now, assume that has a finite presentation =x1,...,xn|R1(x), . . . , Rm(x). Then one considers the connected sum of ncopies of S1×S3, which has then funda- mental group equal to a free group Fn=x1,...,xn. Realizing the relations Rj(x)as loops connecting the base point to mdisjoint embedded circles (i.e., diffeomorphic to S1), one can perform the so called surgery replacing tubular neighbourhoods of these circles, which are diffeomorphic to S1×B3, 123
Topological methods in moduli theory 365 and have boundary S1×S2, each by the manifold with boundary B2×S2. These manifolds are simply connected, hence by van Kampen we introduce the relation Rj(x)=1, and finally one obtains a Mwith fundamental group ∼ =. There are no more restrictions, other than finite presentability, if one requires that be the fundamental group of a compact complex manifold, as shown by Taubes [351], or if one requires that be the fundamental group of a compact symplectic 4-manifold, as shown by Gompf [182]. Notice however that, by the main results of Kodaira’s surface classification (see [21]), the fundamental groups of complex non projective surfaces form a very restricted class, in particular either their Betti number b1is equal to 1, or they sit in an exact sequence of the form 1→Z→→πg→1. In fact Taubes builds on the method of Seifert and Threlfall in order to construct a compact complex manifold Xwith π1(X)=. He takes a (differentiable) 4-manifold Mwith π1(M)=and then, taking the connected sum with a suitable number of copies of the complex projective plane with opposed orientation, he achieves that M allows a metric with anti self dual Weyl tensor. This condition on the metric of M, by a theorem of Atiyah, Hitchin and Singer [14] makes the twistor space Tw(M) a complex manifold (not just an almost complex manifold). Note that (see [268], especially page 366) the twistor space Tw(M)of an oriented Riemannian 4-manifold is an S2-bundle over M, such that the fibre over P∈Mequals the sphere bundle of the rank 3 vector subbundle +⊂2(TM).Here2(TM)=+⊕−is the eigenspace decomposition for the ∗-operator ∗:2(TM)→2(TM), such that ∗2=1. The fact that Tw(M)is an S2-bundle over Mis of course responsible of the iso- morphism π1(Tw(M)) ∼ =π1(M). Returning to Serre’s question, a first result was given by Johnson and Rees [225], which was later extended by other authors [7,196]: Theorem 102 (Johnson–Rees) Let 1, 2be two finitely presented groups admitting some non trivial finite quotient. Then the free product 1∗2cannot be the fundamental group of a normal projective variety. More generally, this holds for any direct product H×(1∗2). The idea of proof is to use the fact that the first cohomology group H1(X,C) carries a nondegenerate skew-symmetric form, obtained simply from the cup product in 1-dimensional cohomology multiplied with the (n−1)-th power of the Kähler class. An obvious observation is that the crucial hypothesis is projectivity. In fact, if we take a quasi-projective variety, then the theorem does not hold true: it suffices to take as Xthe complement of dpoints in a projective curve of genus g, and then π1(X)∼ =F2g+d−1is a free group. This is not just a curve phenomenon: already Zariski showed that if Cis a plane sextic with equation Q2(x)3−G3(X)2=0, where Q2,G3 are generic forms of respective degrees 2,3, then π1(P2\C)∼ =Z/2∗Z/3. One important ingredient for the theorem of Johnson and Rees is the Kurosh sub- group theorem, according to which 1∗2would have a finite index subgroup of the form Z∗K, which would therefore also be the fundamental group of a projective 123
366 F. Catanese variety. Another proof is based on the construction of a flat bundle Ecorresponding to a homomorphism Z∗K→{±1}with some cohomology group Hi(X,E)of odd dimension, contradicting the extension of Hodge theory to flat rank 1 bundles (see [181]). Arapura, Bressler and Ramachandran answer in particular one question raised by Johnson and Rees, namely they show [7]: Theorem 103 (Arapura, Bressler and Ramachandran) If X is a compact Kähler man- ifold, then its universal cover ˜ X has only one end. And π1(X)cannot be a free product amalgamated by a finite subgroup. Here, the ends of a non compact topological space Xare just the limit, as the compact subset Kgets larger, of the connected components of the complement set X\K. The theorem of Johnson and Rees admits the following consequence Corollary 104 (Johnson–Rees) Let X1,X2be smooth projective manifolds of dimen- sion n ≥2with non zero first Betti number, or more generally with the property that π1(Xj)has a non trivial finite quotient. Then X 1!X2cannot be homeomorphic to a projective manifold. It is interesting to observe that the previous result was extended by Donaldson [138] also to the case where X1,X2are simply connected smooth projective surfaces. One could believe that the combination of the two results implies tout court that the con- nected sum X1!X2of two projective manifolds of dimension at least two cannot be homeomorphic to a projective manifold. This is unfortunately not clear (at least to the author), because of a deep result by Toledo [354], giving a negative answer to another question posed by Serre Theorem 105 (Toledo) There exist projective manifolds X whose fundamental group G:= π1(X)is not residually finite. This means that the natural homomorphism of G := π1(X)into its profinite com- pletion ˆ G is not injective π1(X)is also called the algebraic fundamental group and denoted by π1(X)alg. The profinite completion is defined as the following inverse limit ˆ G:= lim K<G,G/K, where K runs through the set of normal subgroups of finite index. Even the following question does not have yet a positive answer. Question 106 Does there exist a projective manifold X such that G =π1(X)= 0, while ˆ G=: π1(X)alg =0?(π1(X)would then be an infinite group). If the answer were negative, then the hypothesis in Corollary 104 would be that X1,X2are not simply connected. 123
Topological methods in moduli theory 367 Concluding this section, it is clear how the results of Siu (and Mostow [300]), can be used to obtain restrictions for Kähler groups. For instance, using the above mentioned techniques of classifying spaces and har- monic maps, plus some Lie theoretic arguments, Carlson and Toledo [75], while giving alternative proofs of the results of Siu, prove: Theorem 107 (Carlson–Toledo) If X is a compact Kähler manifold, then its funda- mental group π1(X)cannot be isomorphic to a discrete subgroup <SO(1,n)such that D/ be compact, where Dis the hyperbolic space S O(1,n)/SO(n). Similar in spirit is the following theorem of Simpson [341], for which we recall that a lattice contained in a Lie group Gis a discrete subgroup such that G/ has finite volume, and that a reductive Lie group is of Hodge type if it has a compact Cartan subgroup. Theorem 108 (Simpson) If X is a compact Kähler manifold, its fundamental group π1(X)cannot be isomorphic to a lattice contained in a simple Lie group not of Hodge type. For more information on Kähler groups (fundamental group of a cKM X), we refer to the book [2], to the survey article by Campana [72] and to [73]. 7.6 Kähler versus projective, Kodaira’s problem and Voisin’s negative answer Many of the restrictions for a group to be a projective group (fundamental group of a projective variety X) are indeed restrictions to be a Kähler group. For long time, it was not clear which was the topological difference between projective smooth varieties and compact Kähler manifolds. Even more, there was a question by Kodaira whether any compact Kähler manifold Xwould be a deformation (or even a direct deformation) of a projective manifold Y, according to the following well known definition. Definition 109 (1) Given two compact complex manifolds Y,X,Yis a direct defor- mation of Xif there is a smooth proper and connected family p:X→B, where Bis a smooth connected complex curve, such that Xand Yisomorphic to some fibres of p. (2) We say instead that Yis deformation equivalent to Xif Yis equivalent to Xfor the equivalence relation generated by the (symmetric) relation of direct deformation. Indeed, Kodaira proved: Theorem 110 (Kodaira [249]) Every compact Kähler surface is a deformation of a projective manifold. It was for a long time suspected that Kodaira’s question, although true in many important cases (tori, surfaces), would be in general false. The counterexamples by Claire Voisin were based on topological ideas, namely on the integrality of some multilinear algebra structures on the cohomology of projective varieties. 123
368 F. Catanese To explain this, recall that a Kähler form ξon a Kähler manifold Xdetermines the Lefschetz operator on forms of type (p,q): L:Ap,q→Ap+1,q+1,L(φ) := ξ∧φ whose adjoint := L∗:Ap,q→Ap−1,q−1 is given by =w∗L∗where w:= p,q (−1)p−qπp,q,(Weil), c:= p,q (i)p−qπp,q,(Chern −Weil), are the Weil and Chern–Weil operators respectively. Here πp,qis the projector onto the space of forms of type (p,q). These operators satisfy the commutation relations: (1) [L,w]=[L,c]=[, w]=[, c]=0 (2) B:= [L,]=(p+q−n)πp,q, (3) [L,B]=−2L. In this way the bigraded algebra of differential forms is a representation of the Lie Algebra sl(2,C), which has the basis: b=−10 01 , l=00 10 , λ=01 00 , and with Lie bracket given by b=[l,λ], −2l=[l,b], 2λ=[λ, b]. For finite dimensional representations of sl(2,C)one has the following structural result. Proposition 111 Let ρ:sl(2,C)−→ End(W)be a representation of sl(2,C), dim(W)<∞, and set L := ρ(l),:= ρ(λ),B:= ρ(b). Then we have: 123
Topological methods in moduli theory 369 (1) W=⊕ ν∈ZWνis a finite direct sum, where Wνis the eigenspace of B with eigenvalue ν. (2) L(Wν)⊂Wν+2,(Wν)⊂Wν−2. (3) Let P ={w|w =0}, be the space of primitive vectors, then we have a direct sum decomposition W=⊕ r∈NLr(P). (4) Moreover, the irreducible representations of sl(2,C)are isomorphic to Sm(C2), where Sm(C2)is the m-th symmetric power of the natural representation of SL(2,C)on C2(i.e., the space of homogeneneous polynomials of degree m). (5) Let Pμ:= P∩Wμ: then Pμ=0for μ>0and P=⊕ μ∈Z,μ≤0Pμ. (6) Lr:P−m−→ W−m+2ris injective for r ≤m and zero for r >m. (7) Lr:W−r−→ Wris an isomorphism ∀r, and Wμ=⊕ r∈N,r≥μLr(Pμ−2r). On a compact Kähler manifold the projectors πp,qcommute with the Laplace operator, so that the components ψp,qof a harmonic form ψare again harmonic, hence the operators L,,... preserve the finite dimensional subspace Hof harmonic forms, and one can apply the above Proposition 111. The following theorem goes often after the name of Lefschetz decomposition, and is essential in order to prove the third Lefschetz theorem (see 3). Theorem 112 (i) Let (X,ξ)be a compact Kähler manifold. Then commutes with the operators ∗,∂,∂,∂∗,∂∗,L,. (ii) In particular, a k- form η=p+q=kηp,qis harmonic if and only if all the ηp,q’s are harmonic. (iii) Hence there is the so-called Hodge Decomposition of the de Rham cohomology: ⊕kHk DR(X,C)∼ =⊕kHk(X,C)=⊕p.qHp,q, where Hp,qis the space of the harmonic Forms of Type (p,q), (iv) Hodge Symmetry holds true, i.e., we have Hp,q=Hq,p. (v) There is a canonical isomorphism Hp,q∼ =Hq(X,p X). (vi) The Lefschetz operators L , make Ha representation of sl(2,C)9, with eigenspaces Hμ:= ⊕p+q=μ+nHp,q, and the space P p,qof primitive forms of type (p,q)allows a decomposition Hp,q=⊕ r∈N,r≥μ=p+q−nLr(Pp−r,q−r), 9Hence also of SL(2,C). 123
370 F. Catanese where P p,q=0for p +q>n. Moreover, the Hermitian product on P p,qgiven by η, η:=in−2pq L(n−p−q)(η ∧η) is positive definite. Now, a projective manifold is a compact Kähler manifold when endowed with the Fubini-Study metric −1 2πi∂∂log|zi|2. This is the Chern class c1(L)where Lis the line bundle OX(1). Kodaira’s embedding theorem characterizes projective manifolds as those compact Kähler manifolds which admit a Kähler metric whose associated form is integral, i.e., it is the first Chern class of a positive line bundle. Definition 113 Let Mbe a compact differentiable manifold of real even dimension dimR(M)=2n. (1) Mis said to admit a real polarized Hodge structure if there exists a real closed two form ξand a Hodge decomposition of its cohomology algebra (i.e., as in iii of Theorem 112) such that the operators L(ψ ) := ξ∧ψ, and :w∗L∗satisfy properties (iv), (v) and (vi) of Theorem 112. (2) Mis said to admit an integral polarized Hodge structure if one can take ξas above in H2(M,Z). Thus the differentiable manifolds underlying a complex Kähler manifolds admit a real polarized Hodge structure, while those underlying a complex projective manifold admit an integral polarized Hodge structure. Voisin showed that these properties can distinguish between Kähler and projective manifolds. We need here perhaps to recall once more that deformation equivalent complex manifolds are diffeomorphic. Theorem 114 (Voisin [364]) In any complex dimension n ≥4there exist compact Kähler manifolds which are not homotopically equivalent to a complex projective manifold. The construction is not so complicated: taking a complex torus Twith an appropriate endomorphism ϕ, and blowing up T×Talong four subtori T×{0},{0}×T, T, ϕ, where Tis the diagonal, and ϕis the graph of ϕ, one obtains the desired Kähler manifold X. The crucial property is the following lemma 123
Topological methods in moduli theory 371 Lemma 115 (Voisin [364]) Assume that the characteristic polynomial f of ϕ, a monic integral polynomial, has all eigenvalues of multiplicity 1, none of them is real, and moreover the Galois group of its splittiing field is the symmetric group S2n. Then T is not an Abelian variety. We refer to the original paper for details of the construction and proof. However, since our Xis obviously bimeromorphic to the torus T×T, which is a deformation of a projective manifold, later Voisin went on to prove a stronger result in [365]. Theorem 116 (Voisin [365]) In any complex dimension n ≥10 there exist compact Kähler manifolds such that no compact smooth bimeromorphic model X of X is homotopically equivalent to a complex projective manifold. Indeed, no such model X has the same rational cohomology ring of a projective manifold. That the rational cohomology ring was the essential topological invariant of Kähler manifolds, had been found already some 30 years before by Deligne, Griffiths, Morgan, and Sullivan [134], using Sullivan’s theory of rational homotopy theory and minimal models of graded differential algebras (see [195] for a thourough introduction, and for some of the notation that we shall introduce here without explanation). Theorem 117 Let X,Y be compact complex manifolds, which are either Kähler, or such that the dd clemma holds for them (as for instance it happens when X is bimero- morphic to a cKM). (1) Then the real homotopy type of X is determined by its real cohomology ring H∗(X,R), and similarly the effect of every holomorphic map f :X→Yon real homotopy types is a formal consequence of the induced homomorphism of real cohomology rings f ∗:H∗(Y,R)→H∗(X,R). (2) If moreover X is simply connected, then the graded Lie algebra of real homotopy groups π∗(X)⊗ZRdepends only on the cohomology ring H ∗(X,R). In particular, all Massey products of any order are zero over Q. (3) If instead X is not simply connected, then the real form of the canonical tower of nilpotent quotients of π1(X)(the real form is obtained by taking ⊗ZRof graded pieces and extension maps) is completely determined by H 1(X,R)and the cup product map H1(X,R)×H1(X,R)→H2(X,R). The article [134] contains several proofs, according to the taste of the several authors. A basic property of compact Kähler manifolds which plays here a key role is the following lemma, also called principle of the two types in [194]. Lemma 118 (ddc-Lemma) Let φbe a differential form such that (1) ∂φ =∂φ =0(equivalently, dφ=dcφ=0, where dcis the real operator −i(∂ −∂)), (2) d(φ) =0(or d c(φ) =0), then 123
372 F. Catanese (3) there exists ψsuch that φ=∂∂ψ (equivalently, there exists ψsuch that φ= ddc(ψ)). One can see the relation of the above lemma with the vanishing of Massey products, which we now define in the simplest case. Definition 119 Let Xbe a topological space, and let a,b,c∈Hi(X,R)be cohomol- ogy classes such that a∪b=b∪c=0, where Ris any ring of coefficients. Then the triple Massey product a,b,cis defined as follows: take cocycle repre- sentatives a,b,cfor a,b,c,respectively, and cochains x,y∈C2i−1(X,R)such that dx =a∪b,dy =b∪c. Then a,b,cis the class of a∪y+(−1)i+1x∪c inside H3i−1(X,R)/(a∪H2i−1(X,R)+H2i−1(X,R)∪c). Concerning the choice of the ring Rof coefficients: if we take Xa cKM, and R=R, or R=Q, then these Massey products vanish, as stated in [134]. Instead things are different for torsion coefficients, as shown by Torsten Ekedahl [150]. Theorem 120 (i) There exist a smooth complex projective surface X and a ,b,c∈ H1(X,Z/l)such that a ∪b=b∪c=0,but a,b,c = 0. (ii) Let X →P6be an embedding and let Y be the blow up of P6along X. Then there exist a,b,c ∈H3(X,Z/l)such that a ∪b =b ∪c =0,but a,b,c = 0. Remark 121 (1) As already mentioned, Theorem 112 is an essential ingredient in the proof of the third Lefschetz theorem. Indeed, for a smooth hyperplane section W=H∩Xof a projective variety X, the cup product with the hyperplane class h∈H2(X,Z)corresponds at the level of forms to the operator Lgiven by wedge product with ξ, the first Chern form of OX(H). (2) Most Kähler manifolds, for instance a general complex torus, do not admit any nontrivial analytic subvariety (nontrivial means: different from Xor from a point). Whereas for projective varieties Xof dimension n≥2 one can take hyperplane sections successively and obtain a surface Swith π1(S)∼ =π1(X). Hence the well known fact that the set of fundamental groups of smooth projective varieties is just the set of fundamental groups of smooth algebraic surfaces. 123
Topological methods in moduli theory 373 (3) As explained in the book by Shafarevich [337], the same idea was used by J.P. Serre to show that any finite group Goccurs as the fundamental group of a smooth projective surface S. Serre considers the regular representation on the Cartesian product of Yindexed by G, which we denote as usual by YG, and where for instance Y=Pmis a simply connected projective variety of dimension m≥3. The quotient X:= (YG)/Gis singular on the image of the big diagonal, which however has codimension at least m. Cutting Xwith an appropriate number of general hyperplanes one obtains a surface Swith π1(S)∼ =G. (4) Serre’s result is used in Ekedahl’s theorem: Ekedahl shows the existence of a group Gof order l5such that there is a principal fibration K(G,1)→K((Z/l)3,1)with fibre a K((Z/l)2,1); moreover he deduces the existence of elements a,b,cas desired from the fact that there is some CW complex with non zero triple Massey product. Then the blow up of P6along a surface Xwith π1(X)∼ =Gis used to obtain an example with classes in the third cohomology group H3(Y,Z/l). 7.7 The Shafarevich conjecture One of the many reasons of the beauty of the theory of curves is given by the uni- formization theorem, stating that any complex manifold Cof dimension 1 which is not of special type (i.e., not P1,C,C∗, or an elliptic curve) has as universal covering the unit disk B1={z∈C||z|<1}, which is biholomorphic to the upper half plane H={z∈C|Im(z)>0}. In other terms, the universal cover is either P1,C,orHaccording to the sign of the curvature of a metric with constant curvature (positive, zero or negative). In higher dimensions there are simply connected projective varieties that have a positive dimensional moduli space: already in the case of surfaces, e.g., smooth surfaces in P3, of degree at least 3, there is an uncountable family of pairwise non isomorphic varieties. So, if there is some analogy, it must be a weaker one, and the first possible direction is to relate somehow Kodaira dimension with curvature. In the case where the canonical divisor KXis ample, there is the theorem of Aubin and Yau (see [16,376]) showing the existence, on a projective manifold with ample canonical divisor KX, of a Kähler–Einstein metric, i.e. a Kähler metric ωsuch that Ric(ω) =−ω. This theorem is partly the right substitute for the uniformization theorem in dimen- sion n>1, but the Kähler–Einstein condition, i.e. the existence of a Kähler metric ω such that Ric(ω) =cω, forces KXample if c<0, KXtrivial if c=0, and −KXample if c>0. 123
374 F. Catanese In the case where KXis trivial, Yau showed the existence of a Kähler–Einstein metric in [376], while the existence of such a metric on Fano manifolds (those with ample anticanonical divisor −KX) has only recently been settled, under a stability assumption (see [122,123,352]). In the general case, one is looking for suitable metrics on singular varieties, but we shall not dwell on this here, since we are focusing on topological aspects: we refer for instance to [162](seealso[204]). Once more, however, curvature influences topology: Yau showed in fact [375] that, for a projective manifold with ample canonical divisor KX, the famous Yau inequality is valid Kn X≤2(n+1) nKn−2 Xc2(X), equality holding if and only if the universal cover Xis the unit ball Bnin Cn. The second possible direction is to investigate properties of the universal covering ˜ Xof a projective variety. For instance, one analogue of projective curves of genus g≥2, whose universal cover is the unit ball B1⊂C, is given by the compact complex manifolds Xwhose universal covering ˜ Xis biholomorphic to a bounded domain ⊂Cn. Necessarily such a manifold Xis projective and has ample canonical divisor KX (see [246,247], Theorem 8.4 p. 144, where the Bergman metric is used, while the method of Poincaré series is used in [340], Theorem 3 p. 117 , see also [255], Chapter 5). Moreover, it is known, by a theorem of Siegel ([339], cf. also [242], Theorem 6.2), that must be holomorphically convex, indeed a Stein domain. We recall these concepts (see for instance [6,185]). Definition 122 A complex space Xis said to be a Stein complex space if and only if one of the following equivalent conditions hold: •(1S) Xis a closed analytic subspace of CN,forsomeN, •(2S) for any coherent analytic sheaf F,wehaveHj(X,F)=0, for all j≥1, •(3S) Xcoincides with the maximal spectrum Specm(HolX)of its algebra of global holomorphic functions HolX:= H0(X,OX), where Specm(HolX)is the sub- space of Spec(HolX)consisting of the maximal ideals. •(4S) Global holomorphic functions separate points and Xpossesses a C∞strictly plurisubharmonic exhaustion function, i.e., there is a real valued proper function f:X→Rsuch that the Levi form L(f):= i∂∂fis strictly positive definite on the Zariski tangent space of each point. A complex space is said to be holomorphically convex if and only if one of the following equivalent conditions hold: •(1HC) For each compact K, its envelope of holomorphy K:= {x∈X|| f(x)|≤ maxK|f|} is also compact •(2HC) Xadmits a proper holomorphic map s:X→to a complex Stein space (this map is called Steinification, or Cartan–Remmert reduction), which induces an isomorphism of function algebras s∗:Hol→HolX. 123
Topological methods in moduli theory 375 •(3HC) Xadmits a real valued twice differentiable proper function f:X→R and a compact K⊂Xsuch that (3i) there is a number c∈Rwith f−1((−∞,c])⊂K (3ii) the Levi form L(f):= i∂∂fis semipositive definite and strictly positive definite outside K. Finally, a complex space Xis Stein manifold iff it is holomorphically convex and global holomorphic functions separate points. Now, a compact complex manifold (or space) is by definition holomorphically convex, so this notion captures two extreme behaviours of the universal covers we just described: bounded domains in Cn, and compact manifolds. In his book [337] Shafarevich answered the following Shafarevich’s question: is the universal covering ˜ Xof a projective variety X,or of a compact Kähler manifold X, holomorphically convex? Remark 123 The answer is negative for a compact complex manifold that is not Kähler, since for instance Hopf surfaces X=(C2\{0})/Zhave C2\{0}as universal cover, which is not Stein since it has the same algebra of holomorphic functions as the larger affine space C2. Observe that all the K(π, 1)projective manifolds we have considered so far satisfy the Shafarevich property, since their universal covering is either Cnor a bounded domain in Cn(this holds also for Kodaira fibred surfaces, by Bers’ simultaneous uniformization, [40]). An interesting question concerns projective varieties whose universal cover is a bounded domain Din Cn. In this case the group Aut (D)contains an infinite cocompact subgroup, so it is natural to look first at domains which have a big group of automor- phisms, especially at bounded homogeneous domains, i.e., bounded domains such that the group Aut() of biholomorphisms of acts transitively. But, as already mentioned, a classical result of J. Hano (see [207] Theorem IV, p. 886, and Lemma 6.2, p. 317 of [292]) asserts that a bounded homogeneous domain that covers a compact complex manifold is a bounded symmetric domain. These naturally occur as such universal covers: Borel proved in [57] that for each bounded symmetric domain there exists a compact free quotient X=/ , called a Hermitian locally symmetric projective manifold (these were also called compact Clifford-Klein forms of the symmetric domain ). By taking finite ramified coverings of such locally symmetric varieties, and blow ups of points of the latter, we easily obtain many examples which are holomorphically convex but not Stein. But if we blow up some subvariety Y⊂Xof positive dimension some care has to be taken: what is the inverse image of Yin the universal cover ˜ X? It depends on the image of π1(Y)inside π1(X): if the image is finite, then we obtain a disjoint union of compact varieties, else the connected components of the inverse image are not compact. A more general discussion leads to the following definition. Definition 124 Assume that the Shafarevich property holds for a compact Kähler manifold: then the fundamental group π1(X)acts properly discontinuously on ˜ X 123
376 F. Catanese and on its Steinification , hence one has a quotient complex space of which is holomorphically dominated by X: Sha f (X):= /π1(X), shafX:X=˜ X/π1(X)→Sha f (X)=/π1(X). Sha f (X)is then called the Shafarevich variety of X, and sha fXis called the Shafarevich morphism. They are characterized by the following universal property: a subvariety Y⊂Xis mapped to a point in Sha f (X)if and only if, letting Yn be the normalization of Y,the image of π1(Yn)→π1(X)is finite. So, a first attempt towards the question was in the 90’s to verify the existence of the Shafarevich morphism shafX. A weaker result was shown by Kollár [254] and by Campana [71]: we borrow here the version of Kóllar, which we find more clearly formulated. Theorem 125 Let X be a normal projective variety: then X admits a rational Sha- farevich map s ha f X:X Shaf (X), with the following properties: (1) shafXhas connected fibres (2) there arecountably many subvarieties Di⊂X such that for every subvariety Y ⊂ X, Y ∪iDi,shaf X(Y)is a point if and only if, letting Y nbe the normalization of Y , the image of π1(Yn)→π1(X)is finite. Moreover, the above properties determine S ha f (X)up to birational equivalence. Remark 126 (1) A subtle point concerning the definition of Shafarevich morphism is the one of passing to the normalization of a subvariety Y. In fact, the normalization Ynhas a holomorphic map to Xsince Xis normal, and the fibre product Yn×X˜ Xhas a holomorphic map to ˜ X. If the image of π1(Yn)→π1(X)is finite, this fibre product consists of a (possibly infinite) Galois unramified covering of Ynwhose components are compact. If ˜ Xis holomorphically convex, then these components map to points in the Steini- fication . (2) Consider the following special situation: Xis a smooth rational surface, and Y is a curve of geometric genus zero with a node. Then the normalization is P1and the Shafarevich map should contract Yn. Assume however that π1(Y)∼ =Zinjects into π1(X): then the inverse image of Yin ˜ Xconsists of an infinite chain of P1’s, which should be contracted by the Steinification morphism, hence this map would not be proper. The existence of a curve satisfying these two properties would then give a negative answer to the Shafarevich question. (3) In other words, the Shafarevich property implies that: (***) if π1(Yn)→π1(X)has finite image , then also π1(Y)→π1(X)has finite image. So, we can define a subvariety to be Shafarevich bad if (SB) π1(Yn)→π1(X)has finite image , but π1(Y)→π1(X)has infinite image. The rough philosophy of the existence of a rational Shafarevich map is thus that Shafarevich bad subvarieties do not move in families which fill the whole X,butthey 123
Topological methods in moduli theory 377 are contained in a countable union of subvarieties Di(one has indeed to take into account also finite union of subvarieties, for instance two P1’s crossing transversally in 2 points: these are called by Kollár ‘normal cycles’). But still, could they exist, giving a negative answer to the Shafarevich question? Potential counterexampes were proposed by Bogomolov and Katzarkov [49], con- sidering fibred surfaces f:X→B. Indeed, for each projective surface, after blowing up a finite number of points, we can always obtain such a fibration. Assume that f:X→Bis a fibration of hyperbolic type: then, passing to a finite unramified covering of Xand passing to a finite covering of the base B,wemay assume that the fibration does not have multiple fibres. Then (see Theorem 96) the surjection π1(X)→π1(B)induces an infinite unram- ified covering ˆ f:ˆ X→ˆ B:= H=B1, and we shall say that this fibration is obtained by opening the base. If instead f:X→Bis a fibration of parabolic type, we get an infinite unramified covering ˆ f:ˆ X→ˆ B:= C. In the elliptic case, B=P1and we set ˆ X:= X,ˆ B:= B. Now, the Shafarevich question has a positive answer if the fundamental group of ˆ X, the image of the fundamental group of a general fibre Finside π1(X), is finite. Assume instead that this image is infinite and look at components of the singular fibres: we are interested to see whether we find some Shafarevich bad cycles. For this, it is important to describe the fundamental group π1(ˆ X)as a quotient of πg=π1(F), where Fis a general fibre, and gis its genus. Observe preliminarily that we may assume that there are singular fibres, else either we have a Kodaira fibration, or an isotrivial fibration (see [285]) so that, denoting by B∗the complement of the critical locus of ˆ f, the fundamental group of B∗is a free group. For each singular fibre Ft,t∈B, a neighbourhood of Ftis retractible to Ft, and because of this we have a continuous map F→Ft. The group of local vanishing cycles is defined as the kernel of π1(F)→ π1(Ft),and denoted by Vant. Then we obtain a description of π1(ˆ X)as π1(ˆ X)=πg/∪t{Vant}, where M denotes the subgroup normally generated by M. Bogomolov and Katzarkov consider the situation where the fibre singularities are exactly nodes, and then for each node pthere is a vanishing cycle van pon the nearby smooth fibre. In this case we divide by the subgroup normally generated by the van- ishing cycles vanp. 123
378 F. Catanese Their first trick is now to replace the original fibration by the pull back via a map ϕ:B→Bwhich is ramified at each critical value tof multiplicity N. In this way they obtain a surface Xwhich is singular, with singular points qof type AN−1, i.e., with local equation zN=xy. Since these are quotient singularities C2/(Z/N), they have local fundamental group π1,loc(X\{q})=Z/N. As a second step, they construct another surface SNsuch that the image of π1(SN) has finite index inside π1(X\Sing(X)). They show (lemma 2.7 and theorem 2.3 of loc. cit.) the following. Proposition 127 The Bogomolov–Katzarkov procedure constructs a new fibred sur- face SNsuch that, if we open the base of the fibration, we get a surface ˆ SNsuch that π1(ˆ SN)=πg/∪p{vanN p}. In this way they propose to construct counterexamples to the Shafarevich ques- tion (with non residually finite fundamental group), provide certain group theoretic questions have an affirmative answer. Remark 128 We want to point out a topological consequence of the Shafarevich prop- erty, for simplicity we consider only the case of a projective surface Xwith infinite fundamental group. Assume that ˜ Xis holomorphically convex, and let s:˜ X→the Cartan–Remmert reduction morphism, where is Stein and simply connected. If dim() =1,then is contractible, otherwise we know [4] that is homotopy equivalent to a CW complex of real dimension 2. In the first case we have a fibration with compact fibres of complex dimension 1, in the second case we have a discrete set such that the fibre has complex dimension 1, and the conclusion is that: if a projective surface with infinite fundamental group satisfies the Shafarevich property, then ˜ Xis homotopy equivalent to a CW complex of real dimension at most 2. To my knowledge, even this topological corollary of the Shafarevich property is yet unproven. On the other hand, the main assertion of the Shafarevich property is that the quo- tient of ˜ Xby the equivalence relation which contracts to points the compact analytic subspaces of ˜ Xis not only a complex space, but a Stein space. This means that one has to produce a lot of holomorphic functions on ˜ X, in order to embed the quotient as a closed analytic subspace in some CN. There are positive results, which answer the Shafarevich question in affirmative provided the fundamental group of Xsatisfies certain properties, related somehow to some of the themes treated in this article, which is the existence of certain homomor- phisms to fundamental groups of classifying spaces, and to the theory of harmonic maps. To give a very very simple idea, an easy result in this directions is that ˜ Xis a Stein manifold if the Albanese morphism α:X→A:= Alb(X)is a finite covering. 123
Topological methods in moduli theory 379 Now, since there has been a series of results in this direction,we refer to the intro- duction and bibliography in the most recent one [160] for some history of the problem and for more information concerning previous results, especially the first ones due to Jost and Zuo (for instance, [198,229–231]) and other ones. We want however to directly cite some previous results due to Katzarkov and Ramachandran [234], respectively to Eyssidieux [159](seealso[161] for a general introduction) which are simpler to state. We only want to recall that a reductive rep- resentation is one whose image group is reductive, i.e., all of its representations are semi-simple, or completely reducible (any invariant subspace W⊂Vhas an invariant complement). Theorem 129 (Katzarkov–Ramachandran) Let X be a normal Kähler compact sur- face, and X →X an unramified cover with Galois group (so X =X/).If does not contain Zas a finite index subgroup and it admits an almost faithful (i.e., with finite kernel)Zariski dense representation in a connected reductive complex Lie group, then X is holomorphically convex. Theorem 130 (Eyssidieux) Let X be a smooth projective variety, consider a homo- morphism ρ:π1(X)→G, and let ˜ Xρ:= ˜ X/ker(ρ) be the connected unramified covering of X associated to k er (ρ). (1) If G =GL(n,C)and ρis a reductive representation, then there is a relative Shafarevich morphism shafρ:X→Sha fρ(X) to a normal projective variety, satisfying the universal property that for each normal variety Z mapping to X , its image in Sha fρ(X)is a point if and only if the image of π1(Z)→π1(X)is finite. Moreover, the connected components of the fibres of ˜ Xρ→Shafρ(X)are compact. (2) Let M be a quasi compact and absolute constructible (i.e., it remains constructible after the action of each σ∈Aut (C))set of conjugacy classes of reductive linear representations of π1(X), and let HMbe the intersection of the respective kernels. Then ˜ XM:= ˜ X/HMis holomorphically convex. The main result of [160] consists of two parts, the first one being the following: Theorem 131 (Eyssidieux–Katzarkov–Pantev–Ramachandran) Let X be a smooth projective variety, and let G be a reductive algebraic group defined over Q. Consider the Betti character scheme M := MB(X,B)such that, for a C-algebra A of finite type, the set of A-valued points M(A)=M(Spec(A)) parametrizes the representations (∗)ρ:π1(X)→G(A). (1) Denote by ˜ H∞ Mthe kernel of all such representations: then the associated Galois connected unramified covering ˜ X∞ Mis holomorphically convex. 123
380 F. Catanese (2) there is a non-increasing family of normal subgroups ˜ Hk M(obviously containing ˜ H∞ M), which correspond to homomorphisms ρ:π1(X)→G(A), with the prop- erty that A is an Artinian local C-algebra, and that the Zariski closure of the image has k-step unipotent radical. Then the associated Galois connected unramified covering ˜ Xk Mis holomorphically convex. 7.8 Strong and weak rigidity for projective K(π, 1)manifolds We want here to see how the moduli problem is solved for many projective K(π, 1) manifolds, which were mentioned in Sect. 3. However, we want here not to be entan- gled in the technical discussion whether a variety of moduli exists, and, if so, which properties does it have. So, we use the notion of deformation equivalence introduced by Kodaira and Spencer. Definition 132 (Rigidity) (1) Let Xbe a projective manifold: then we say that Xis strongly rigid if, for each projective manifold Yhomotopy equivalent to X, then Yis isomorphic to Xor to the complex conjugate variety ¯ X(X∼ =Yor ¯ X∼ =Y). (2) We say instead that Xis weakly rigid if, for each other projective manifold Y homotopy equivalent to X, then either Yis direct deformation of X,orYis direct deformation of ¯ X(see 109). (3) We say instead that Xis quasi rigid if, for each other projective manifold Y homotopy equivalent to X, then either Yis deformation equivalent to X,or Yis deformation equivalent to ¯ X(recall that deformation equivalence is the equivalence relation generated by the relation of direct deformation). Remark 133 (a) This is the intuitive meaning of the above definition: assume that there is a moduli space M, whose points correspond to isomorphism classes of certain varieties, and such that for each flat family p:X→Bthe natural map B→Massociating to b∈Bthe isomorphism class of the fibre Xb:= p−1(b) is holomorphic. Then to be a direct deformation of each other means to belong to the same irreducible component of the moduli space M, while being deformation equivalent means to belong to the same connected component of the moduli space M. (b) The same definition can be given in the category of compact Kähler manifolds, or in the category of compact complex manifolds. (c) Of course, if p:X→Bis a smooth proper family with base Ba smooth con- nected complex manifold, then all the fibres Xbare direct deformation equivalent. (d) We shall need in the sequel some more technical generalization of these notions, which amount, given a product of varieties X1×X2× ···× Xh, to take the complex conjugate of a certain number of factors. Definition 134 Let Xbe a projective manifold: then we say that Xis strongly * rigid if, for each other projective manifold Yhomotopy equivalent to X, then Yis 123
Topological methods in moduli theory 381 isomorphic to Xor to another projective variety Zobtained by Xas follows: there is a Galois unramified covering Xof X( thus X=X/G) which splits as a product X=X 1×X 2×···×X h. Set then, after having fixed a subset J⊂{1,...,h}:Zj:= Xjfor j∈J, and Zj:= Xjfor j/∈J. Consider then the action of Gon Z=(Z 1×Z 2×···×Z h), and let Z:= Z/G. Replacing the word: ‘isomorphic’ by ‘direct deformation equivalent’, resp. ‘defor- mation equivalent’, we define the notion of weakly * rigid, resp. quasi * rigid. (1) The first example, the one of projective curves, was essentially already discussed: we have the universal family pg:Cg→Tg, hence, according to the above definition, projective curves are weakly rigid (and in a stronger way, since we do not need to allow for complex conjugation, as the complex conjugate ¯ Cis a direct deformation of C). (2) Complex tori are weakly rigid in the category of compact Kähler manifolds, since any manifold with the same integral cohomology of a complex torus is a complex torus (Theorem 77). Moreover, complex tori are parametrized by an open set Tnof the complex Grass- mann Manifold Gr(n,2n), image of the open set of matrices {∈Mat(2n,n;C)| indet() > 0},as follows: we consider a fixed lattice ∼ =Z2n, and to each matrix as above we associate the subspace V=Cn, so that V∈Gr(n,2n)and ⊗C∼ =V⊕¯ V. Finally, to we associate the torus YV:= V/pV(),pV:V⊕¯ V→Vbeing the projection onto the first addendum. As it was shown in [100](cf.also[103]) Tnis a connected component of Teichmüller space. (3) Complex tori are not quasi rigid in the category of compact complex manifolds. Sommese generalized some construction by Blanchard and Calabi, obtaining [348] that the space of complex structures on a six dimensional real torus is not connected. (4) Abelian varieties are quasi rigid, but not weakly rigid. In fact, we saw that all the Abelian varieties of dimension gadmitting a polarization of type Dare contained in a family over Hg. Moreover, products E1×···×Egof elliptic curves admit polarizations of each possible type. (5) Locally symmetric manifolds D/ where Dis irreducible of dimension >1are strongly rigid by Siu’s theorem 90. 123
382 F. Catanese (6) Locally symmetric manifolds D/ with the property that Ddoes not have any irreducible factor of dimension 1 are strongly * rigid by Siu’s theorem 90. (7) Varieties isogenous to a product (VIP) are weakly * rigid in all dimensions, according to Theorem 136, that we are going to state soon (see below). They are weakly rigid in dimension n=2 only if we require the homotopy equivalence to be orientation preserving. (8) Among the VIP’s, the strongly * rigid are exactly the quotients X=(C1×C2× ···×Cn)/Gwhere Gnot only acts freely, but satisfies the following property. Denote by G0⊂G(see [96] for more details) the subgroup which does not permute the factors, and observe that G0⊂Aut(C1)×Aut(C2)×···×Aut(Cn). Then G0operates on each curve Ci, and the required condition is that this action is rigid, more precisely we want : Ci/G0∼ =P1and the quotient map pi:Ci→ Ci/G0∼ =P1is branched in three points (we shall also say that we have then a triangle curve). Again, in dimension n=2 we get some strongly rigid surfaces, which have been called (ibidem) Beauville surfaces. We propose therefore to call Beauville varieties the strongly * rigid VIP’s. (9) Hyperelliptic surfaces are weakly rigid, as classically known, see below for a more general result. (10) It is unclear, as we shall see, whether Kodaira fibred surfaces are weakly * rigid, however any surface homotopically equivalent to a Kodaira fibred surface is also a Kodaira fibred surface. Indeed a stronger result, with a similar method to the one of [96] was shown by Kotschick (see [258] and Theorem 138 below), after some partial results by Jost and Yau. (11) Generalized hyperelliptic varieties Xare a class of varieties for which a weaker property holds. Namely, if Yis a compact Kähler manifold which is homotopy equivalent to X(one can relax this assumption a bit, obtaining stronger results), then Yis the quotient of Cnby an affine action of := π1(X)∼ =π1(Y)which, using Proposition 150, can be shown to have the same real affine type as the action yielding Xas a quotient. But the Hodge type could be different, except in special cases where weak rigidity holds. The following (see [96,98]) is the main result concerning surfaces isogenous to a product, and is a stronger result than weak * rigidity. Theorem 135 (a) A projective smooth surface S is isogenous to a product of two curves of respective genera g1,g2≥2, if and only if the following two conditions are satisfied: (1) there is an exact sequence 1→πg1×πg2→π=π1(S)→G→1, where G is a finite group and where πgidenotes the fundamental group of a projective curve of genus gi≥2; 123
Topological methods in moduli theory 383 (2) e(S)(=c2(S)) =4 |G|(g1−1)(g2−1). (b) Write S =(C1×C2)/ G. Any surface X with the same topological Euler number and the same fundamental group as S is diffeomorphic to S and is also isogenous to a product. There is a smooth proper family with connected smooth base manifold T , p :X→T having two fibres respectively isomorphic to X , and Y , where Y is one of the 4 surfaces S =(C1×C2)/G, S+− := (C1×C2)/G, ¯ S=(C1×C2)/G, S−+ := (C1×C2)/G=S+−. (c) The corresponding subset of the moduli space of surfaces of general type Mtop S=Mdi f f S, corresponding to surfaces orientedly homeomorphic, resp. orientedly diffeomorphic to S, is either irreducible and connected or it contains two connected components which are exchanged by complex conjugation. In particular, if S is orientedly diffeomorphic to S, then Sis deformation equivalent to S or to ¯ S. Idea of the proof := π1(S)admits a subgroup of index dsuch that ∼ = (πg1×πg2).LetSbe the associated unramified covering of S. Then application of the isotropic subspace theorem or of Theorem 96 yields a pair of holomorphic maps fj:S→Cj, hence a holomorphic map F:= f1×f2:S→C 1×C 2. Then the fibres of f1have genus h2≥g2, hence by the Zeuthen Segre formula (2) e(S)≥4(h2−1)(g1−1), equality holding if and only if all the fibres are smooth. But e(S)=4(g1−1)(g2−1)≤4(h2−1)(g1−1),soh2=g2, all the fibres are smooth hence isomorphic to C 2; therefore Fis an isomorphism. The second assertion follows from the refined Nielsen realization theorem, The- orem 37, considering the action of the index two subgroup G0of G. Notice that if a group, here G0, acts on a curve C, then it also acts on the complex conjugate curve, thus G0⊂Aut(C)⇒G0⊂Aut(¯ C). But since complex conjugation is ori- entation reversing, these two actions are conjugate by Out(πg), not necessarily by Mapg=Out+(πg). Recall now that the exact sequence 1→πg1×πg2→0→G0→1 yields two injective homomorphisms ρj:G0→Out(πgj)for j=1,2. Now, we choose the isomorphism π1(Cj)∼ =πgjin such a way that it is orientation preserving. The isomorphism π1(X)∼ =π1(S), and the realization X=(C 1×C 2)/Gthen gives an isomorphism φj:π1(Cj)∼ =πgj. If this isomorphism is orientation preserving, then the two actions of G0on Cj,Cjare deformation of each other by the refined Nielsen realization. If instead this isomorphism is orientation reversing, then we replace Cj by Cjand now the two actions are conjugate by Mapg, so we can apply the refined Nielsen realisation theorem to Cj,Cj. Finally, in the case where the homeomorphism of Xwith Sis orientation preserving, then either both φ1,φ 2are orientation preserving, or they are both orientation reversing. Then Xis a deformation either of Sor of ¯ S. 123
384 F. Catanese The first part of the following theorem is instead a small improvement of theorem 7.1 of [96], the second part is theorem 7.7 ibidem, and relies on the results of Mok [294, 295]. Theorem 136 (1) Let Y be a projective variety of dimension n with K Yample and that := π1(Y)admits a subgroup of index d such that ∼ =πg1×πg2×···×πgn,gi≥2∀i, and moreover H 2n(, Z)→H2n(Y,Z)is an isomorphism. Then Y is a variety isogenous to a product. (2) Let X be projective variety with universal covering the polydisk Hn, and let Y be a projective variety of dimension n with K Yample and such that π1(Y)∼ = := π1(X), and assume that H 2n(, Z)→H2n(Y,Z)is an isomorphism. Then also Y has Hnas universal covering (hence we have a representation of Y as a quotient Y =Hn/). (3) In particular X is weakly * rigid, and strongly * rigid if the action of is irreducible (this means that there is no finite index subgroup <and an isomorphism Hn∼ =Hm×Hn−msuch that = 1× 2, 1⊂Aut(Hm), 2⊂Aut(Hn−m). Proof of (1): As in the previous theorem we take the associated unramified covering Yof Yassociated to , and with the same argument we produce a holomorphic map F:Y→Z:= C1×C2×···×Cn. We claim that Fis an isomorphism. Indeed, by the assumption H2n(, Z)∼ = H2n(Y,Z)follows that H2n(,Z)∼ =H2n(Y,Z), hence Fhas degree 1 and is a birational morphism. Let Rbe the ramification divisor of F, so that KY=F∗(KZ)+ R. Now, since Fis birational we have H0(mKY)∼ =H0(mKZ), for all m≥0. This implies for the corresponding linear systems of divisors: |mKY|=F∗(|mKZ|)+mR. For m>> 0 this shows that KYcannot be ample, contradicting the ampleness of KY. Remark 137 Part (2) of the above theorem is shown in [96], while part (3) follows by the same argument given in the proof of Theorem 135 in the case where Xis isogenous to a product. We omit the proof of (3) in general. Theorem 138 (Kotschick) Assume that S is a compact Kähler surface, and that (i) its fundamental group sits into an exact sequence, where g,b≥2: 1→πg→π1(S)→πb→1 123
Topological methods in moduli theory 385 (ii) e(S)=4(b−1)(g−1). Then S has a smooth holomorphic fibration f :S→B , where B is a projective curve of genus b, and where all the fibres are smooth projective curves of genus g. f is a Kodaira fibration if and only if the associated homomorphism ρ:πb→Mapghas image of infinite order, else it is a surface isogenous to a product of unmixed type and where the action on the first curve is free. Proof By Theorem 96 the above exact sequence yields a fibration f:S→Bsuch that there is a surjection π1(F)→πg, where Fis a smooth fibre. Hence, denoting by hthe genus of F, we conclude that h≥g, and again we can use the Zeuthen–Segre formula to conclude that h=gand that all fibres are smooth. So Fis a smooth fibration. Let C→Cbe the unramified covering associated to ker(ρ): then the pull back family S→Chas a topological trivialization, hence is a pull back of the universal family Cg→Tgfor an appropriate holomorphic map ϕ:C→Tg. If ker(ρ) has finite index, then Cis compact and, since Teichmüller space is a bounded domain in C3g−3, the holomorphic map is constant. Therefore Sis a product C×C2and, denoting by G:= Im(ρ),S=(C×C2), and we get exactly the surfaces isogenous to a product such that the action of Gon the curve Cis free. If instead G:= Im(ρ) is infinite, then the map of Cinto Teichmüller space is not constant, since the isotropy group of a point corresponding to a curve Fis, as we saw, equal to the group of automorphisms of F(which is finite). Therefore, in this case, we have a Kodaira fibration. Remark 139 Jost and Yau [226] proved a weaker result, that a deformation of the original examples by Kodaira of Kodaira fibrations are again Kodaira fibrations. In our joint paper with Rollenske [109] we gave examples of Kodaira fibred surfaces which are rigid, and we also described the irreducible connected components of the moduli space corresponding to the subclass of those surfaces which admit two different Kodaira fibrations. Question 140 Are Kodaira fibred surfaces weakly * rigid? Let us explain why one can ask this question: a Kodaira fibration f:S→B yields a holomorphic map ϕ:B→Mg. The unramified covering Bassociated to ker(ρ ),ρ:π1(B)→Mapgadmits then a holomorphic map ϕ:B→Tgwhich lifts ϕ.IfMgwere a classifying space the homomorphism ρwould determine the homotopy class of ϕ. A generalization of the cited theorems of Eells and Samson and of Hartmann ([147,209], see theorem 73)would then show that in each class there is a unique harmonic representative, thus proving that, fixed the complex structure on B, there is a unique holomorphic representative, if any. This might help to describe the locus thus obtained in Tb(the set of maps ϕ:B→Tgthat are ρ-equivariant). But indeed, our knowledge of Kodaira fibrations is still scanty, for instance the following questions are still open. Question 141 (1) Given an exact sequence 1→πg→π→πb→1where g ≥3, b≥2, does there exist a Kodaira fibred surface S with fundamental group π? (2) (Le Brun’s question, see [109]) : are there Kodaira fibred surfaces with slope c2 1/c2>8/3? 123
386 F. Catanese (3) Are there surfaces admitting three different Kodaira fibrations? A brief comment on question (1) above. The given exact sequence determines, via conjugation of lifts, a homomorphism ρ:πb→Out+(πg)=Mapg. If the image G of ρis finite, then πis the fundamental group of a surface isogenous to a product, and the answer is positive, as we previously explained. Case (II) : ρis injective, hence (as πbhas no nontrivial elements of finite order) G=Im(ρ) acts freely on Teichmüller space Tg, and M:= Tg/Gis a classifying space for πb. There is a differentiable map of a curve of genus binto M, and the question is whether one can deform it to obtain a harmonic or holomorphic map. Case (III) : ρis not injective. This case is the most frequent one, since, given a Kodaira fibred surface f:S→B, for each surjection F:B→B, the pull-back S:= S×BBis again a Kodaira fibration and ρfactors as the composition of the surjection π1(B)→π1(B)with ρ:π1(B)→Mapg. It is interesting to observe, via an elementary calculation, that the slope K2 S/e(S)> 2 can only decrease for a branched covering B→B, and that it tends to 2 if moreover the weight of the branch divisor β:= (1−1/mi)tends to ∞. Similar phenomena (ker(ρ) = 0, and the slope decreases) occur if we take general hypersurface sections of large degree B⊂M∗ gin the Satake compactification of the moduli space Mgwhich intersect neither the boundary ∂M∗ g=M∗ g\Mgnor the locus of curves with automorphisms (both have codimension at least 2). 7.9 Can we work with locally symmetric varieties? As we shall see in the sequel, Abelian varieties and curves are simpler objects to work with, because explicit constructions may be performed via bilinear algebra in the former case, and via ramified coverings in the latter. For locally symmetric varieties, the constructions are more difficult, hence rigidity results are not enough to decide in concrete cases whether a given variety is locally symmetric. Recently, as a consequence of the theorem of Aubin and Yau on the existence of a Kähler–Einstein metric on projective varieties with ample canonical divisor KX, there have been several simple explicit criteria which guarantee that a projective variety with ample canonical divisor KXis locally symmetric (see [117,118,362,377]). A difficult question which remains open is the one of studying actions of a finite group Gon them, and especially of describing the G-invariant divisors. 8 Inoue type varieties While a couple of hundreds examples are known today of families of minimal surfaces of general type with geometric genus pg(S):= dim H0(OS(KS)) =0 (observe that for these surfaces 1 ≤K2 S≤9), for the value K2 S=7 there are only two examples known (cf. [121,222]), and for a long time only one family of such surfaces was known, the one constructed by Masahisa Inoue (cf. [222]). The attempt to prove that Inoue surfaces form a connected component of the moduli space of surfaces of general type proved to be successful [35], and was based on a 123
Topological methods in moduli theory 387 weak rigidity result: the topological type of an Inoue surface determines an irreducible connected component of the moduli space (a phenomenon similar to the one which was observed in several papers, as [30,31,36,119]). The starting point was the calculation of the fundamental group of an Inoue surface with pg=0 and K2 S=7: it sits in an extension (πgbeing as usual the fundamental group of a projective curve of genus g): 1→π5×Z4→π1(S)→(Z/2Z)5→1. This extension is given geometrically, i.e., stems from the observation [35] that an Inoue surface Sadmits an unramified (Z/2Z)5—Galois covering ˆ Swhichisanample divisor in E1×E2×D, where E1,E2are elliptic curves and Dis a projective curve of genus 5; while Inoue described ˆ Sas a complete intersection of two non ample divisors in the product E1×E2×E3×E4of four elliptic curves. It turned out that the ideas needed to treat this special family of Inoue surfaces could be put in a rather general framework, valid in all dimensions, setting then the stage for the investigation and search for a new class of varieties, which we proposed to call Inoue-type varieties. Definition 142 ([35]) Define a complex projective manifold Xto be an Inoue-type manifold if (1) dim(X)≥2; (2) there is a finite group Gand an unramified G-covering ˆ X→X, (hence X=ˆ X/G) such that (3) ˆ Xis an ample divisor inside a K(, 1)-projective manifold Z, (hence by the theorems of Lefschetz, see Theorem 3,π1(ˆ X)∼ =π1(Z)∼ =) and moreover (4) the action of Gon ˆ Xyields a faithful action on π1(ˆ X)∼ =: in other words the exact sequence 1→∼ =π1(ˆ X)→π1(X)→G→1 gives an injection G→Out(), defined by conjugation by lifts of elements of G; (5) the action of Gon ˆ Xis induced by an action of Gon Z. Similarly one defines the notion of an Inoue-type variety, by requiring the same properties for a variety Xwith canonical singularities. Example 143 Indeed, the examples of Inoue, which also allow him to find another description of the surfaces constructed by Burniat [65], are based on products of curves, and there the group Gis a Z/2-vector space. In fact things can be done more algebraically (as in [36]). If we take a (Z/2)3- covering of P1branched on 4 points, it has equations in P3: x2 1+x2 2+x2 3=0,x2 0−a2x2 2−a3x2 3=0, the group G∼ =(Z/2)3is the group of transformations xi→±xi, the quotient is P1=y∈P3|y1+y2+y3=0,y0−a2y2−a3y3=0, 123
388 F. Catanese and the branch points are the 4 points yi=0,i=0,1,2,3. The group G, if we see the elliptic curve as C/(Z+Zτ), is the group of affine transformations [z] →±[z]+1 2(a+bτ), a,b∈{0,1}, with linear coefficient ±1 and translation vector a point of 2-torsion. Algebra becomes easier than theta functions if one takes several square (or cubic) roots: for instance 3 square roots define a curve of genus 5 in P4: x2 1+x2 2+x2 3=0,x2 0−a2x2 2−a3x2 3=0,x2 4−b2x2 2−b3x2 3=0. One can take more generally a (Z/2)ncovering of P1branched on n+1 points, which is a curve of genus g=2n−2(n−3)+1, or curves corresponding to similar Kummer coverings ((Z/m)ncoverings of P1branched on n+1 points). The above definition of Inoue type manifold, although imposing a strong restriction on X, is too general, and in order to get weak rigidity type results it is convenient to impose restrictions on the fundamental group of Z, for instance the most interesting case is the one where Zis a product of Abelian varieties, curves, and other locally symmetric varieties with ample canonical bundle. Definition 144 We shall say that an Inoue-type manifold Xis (1) a special Inoue type manifold if moreover Z=(A1×···×Ar)×(C1×···×Ch)×(M1×···×Ms) where each Aiis an Abelian variety, each Cjis a curve of genus gj≥2, and Mi is a compact quotient of an irreducible bounded symmetric domain of dimension at least 2 by a torsion free subgroup; (2) a classical Inoue type manifold if moreover Z=(A1×···×Ar)×(C1×···×Ch) where each Aiis an Abelian variety, each Cjis a curve of genus gj≥2; (3) a special Inoue type manifold is said to be diagonal if moreover: (I) the action of Gon ˆ Xis induced by a diagonal action on Z, i.e., G⊂ r i=1 Aut(Ai)× h j=1 Aut(Cj)× s l=1 Aut(Ml)(145) and furthermore: (II) the faithful action on π1(ˆ X)∼ =, induced by conjugation by lifts of elements of Gin the exact sequence 1→=r i=1(i)×h j=1(πgj)×s l=1(π1(Ml)) →π1(X)→G→1 (146) 123
Topological methods in moduli theory 389 (observe that each factor i, resp. πgj,π 1(Ml)is a normal subgroup), satisfies the Schur property (SP)Hom(Vi,Vj)G=0,∀i= j. Here Vj:= j⊗Qand, in order that the Schur property holds, it suffices for instance to verify that for each ithere is a subgroup Hiof Gfor which Hom(Vi,Vj)Hi=0,∀j= i. The Schur property (SP) plays an important role in order to show that an Abelian variety with such a G-action on its fundamental group must split as a product. Before stating the main general result of [35] we need the following definition, which was already used in 77 for the characterization of complex tori among Kähler manifolds. Definition 147 Let Y,Ybe two projective manifolds with isomorphic fundamental groups. We identify the respective fundamental groups π1(Y)=π1(Y)=. Then we say that the condition (SAME HOMOLOGY) is satisfied for Yand Yif there is an isomorphism ":H∗(Y,Z)∼ =H∗(Y,Z)of homology groups which is compatible with the homomorphisms u:H∗(Y,Z)→H∗(, Z), u:H∗(Y,Z)→H∗(, Z), i.e., "satisfies u◦"=u. We can now state the following Theorem 148 Let X be a diagonal special Inoue type manifold, and let Xbe a projective manifold with K Xnef and with the same fundamental group as X , which moreover either (A) is homotopically equivalent to X; or satisfies the following weaker property: (B) let ˆ Xbe the corresponding unramified covering of X. Then ˆ X and ˆ Xsatisfy the condition (SAME HOMOLOGY). Setting W := ˆ X, we have that (1) X=W/G where W admits a generically finite morphism f :W→Z, and where Zis also a K (, 1)projective manifold, of the form Z =(A 1×···×A r)× (C 1×···×C h)×(M 1×···×M s). Moreover here M iis either Mior its complex conjugate, and the product decom- position corresponds to the product decomposition (146)of the fundamental group of Z. The image cohomology class f∗([W])corresponds, up to sign, to the cohomology class of ˆ X. (2) The morphism f is finite if n =dim X is odd, and it is generically injective if (**)the cohomology class of ˆ X(in H∗(Z,Z))is indivisible, or if every strictly sub- multiple cohomology class cannot be represented by an effective G-invariant divisor on any pair (Z,G)homotopically equivalent to (Z,G). 123
390 F. Catanese (3) f is an embedding if moreover K Xis ample, (*)every such divisor W of Zis ample, and (***)Kn X=Kn X.10 In particular, if K Xis ample and (*),(**)and (***)hold, also X is a diagonal SIT (special Inoue type)manifold. A similar conclusion holds under the alternative assumption that the homotopy equivalence sends the canonical class of W to that of ˆ X : then X is a minimal resolution of a diagonal SIT (special Inoue type) variety. Hypothesis (A) in Theorem 148 is used to derive the conclusion that also W:= ˆ X admits a holomorphic map fto a complex manifold Zwith the same structure as Z, while hypotheses (B) and following ensure that the morphism is birational onto its image, and the class of the image divisor f(ˆ X)corresponds to ±that of ˆ Xunder the identification H∗(Z,Z)∼ =H∗(, Z)∼ =H∗(Z,Z). Since KXis ample, one uses (***) to conclude that fis an isomorphism with its image. The next question which the theorem leaves open is weak * rigidity, for which several ingredients should come into play: the Hodge type, a fine analysis of the structure of the action of Gon Z, the problem of existence of hypersurfaces on which G acts freely and the study of the family of such invariant effective divisors, in particular whether the family has a connected base. We have restricted ourselves to special Inoue type manifolds in order to be able to use the regularity results for classifying maps discussed in the previous section (the diagonality assumption is only a simplifying assumption). Let us now sketch the proof of Theorem 148. Proof of Theorem 148 Step 1 The first step consists in showing that W:= ˆ Xadmits a holomorphic mapping to a manifold Zof the above type Z=(A 1×···×A r)× (C 1×···×C h)×(M 1×···×M s), where M iis either Mior its complex conjugate. First of all, by the results of Siu and others ([101,106,344,345], Theorem 5.14) cited in Sect. 6,Wadmits a holomorphic map to a product manifold of the desired type Z 2×Z 3=(C 1×···×C h)×(M 1×···×M s). Look now at the Albanese variety Alb(W)of the Kähler manifold W, whose fun- damental group is the quotient of the Abelianization of =π1(Z)by its torsion subgroup. Write the fundamental group of Alb(W)as the first homology group of A×Z2×Z3, i.e., as H1(Alb(W)) =⊕H1(Z2,Z)⊕(H1(Z3,Z)/Torsion), 10 This last property for algebraic surfaces follows automatically from homotopy invariance. 123
Topological methods in moduli theory 391 (Alb(Z2)is the product of Jacobians J:= (Jac(C1)×···×Jac(Ch))). Since however, by the universal property, Alb(W)has a holomorphic map to B:= Alb(Z 2)×Alb(Z 3), inducing a splitting of the lattice H1(Alb(W), Z)=⊕H1(B,Z),itfollowsthat Alb(W)splits as A×B. Now, we want to show that the Abelian variety A(Wis assumed to be a pro- jective manifold) splits as desired. This is in turn a consequence of assumption (3) in Definition 144. In fact, the group Gacts on the Abelian variety Aas a group of biholomorphisms, hence it acts on ⊗Rcommuting with multiplication by √−1. Hence multiplication by √−1 is an isomorphism of Grepresentations, and then (3) implies that i⊗Ris stable by multiplication by √−1; whence i⊗Rgenerates a subtorus A i. Finally, Asplits because is the direct sum of the sublattices i.We are through with the proof of step 1. Step 2 Consider now the holomorphic map f:W→Z. We shall show that the image W:= f(W)is indeed a divisor in Z. For this we use the assumption (SAME HOMOLOGY) and, in fact, the claim is an immediate consequence of the following lemma. Lemma 149 Assume that W is a Kähler manifold, such that (i) there is an isomorphism of fundamental groups π1(W)=π1(ˆ X)=, where ˆ X is a smooth ample divisor in a K (, 1)complex projective manifold Z ; (ii) there exists a holomorphic map f :W→Z, where Z is another K (, 1) complex manifold, such that f∗:π1(W)→π1(Z)=is an isomorphism, and moreover (iii) (SAME HOMOLOGY) there is an isomorphism ":H∗(W,Z)∼ =H∗(ˆ X,Z) of homology groups which is compatible with the homomorphisms u:H∗(ˆ X,Z)→H∗(, Z), u:H∗(W,Z)→H∗(, Z), i.e., we have u ◦"=u. Then f is a generically finite morphism of W into Z , and the cohomology class f∗([W])in H∗(Z,Z)=H∗(Z,Z)=H∗(, Z) corresponds to ±1the one of ˆ X. Proof of the Lemma We can identify the homology groups of Wand ˆ Xunder ": H∗(W,Z)∼ =H∗(ˆ X,Z), and then the image in the homology groups of H∗(Z,Z)= H∗(Z,Z)=H∗(, Z)is the same. We apply the above consideration to the fundamental classes of the oriented mani- folds Wand ˆ X, which are generators of the infinite cyclic top degree homology groups H2n(W,Z), respectively H2n(ˆ X,Z). 123
392 F. Catanese This implies a fortiori that f:W→Zis generically finite: since then the homol- ogy class f∗([W])(which we identify to a cohomology class by virtue of Poincaré duality) equals the class of ˆ X, up to sign. Step 3 We claim that f:W→Zis generically 1-1 onto its image W. Let dbe the degree of f:W→W. Then f∗([W])=d[W], hence if the class of ˆ Xis indivisible, then obviously d=1. Otherwise, observe that the divisor Wis an effective G-invariant divisor and use our assumption (**). Step 4 Here we are going to prove that fis an embedding under the additional hypotheses that Kn X=Kn Xand that Wis ample, as well as KX. We established that fis birational onto its image W, hence it is a desingularization of W. We use now adjunction. We claim that, since (by our assumption on KX)KWis nef, there exists an effective divisor U, called the adjunction divisor, such that KW=f∗(KZ+W)−U. This can be shown by taking the Stein factorization p◦h:W→WN→W, where WNis the normalization of W. Let Cbe the conductor ideal Hom(p∗OWN,OW)viewed as an ideal C⊂OWN; then the Zariski canonical divisor of WNsatisfies KWN=p∗(KW)−C=p∗(KZ+W)−C where Cis the Weil divisor associated to the conductor ideal (the equality on the Gorenstein locus of WNis shown for instance in [83], then it suffices to take the direct image from the open set to the whole of WN). In turn, we would have in general KW=h∗(KWN)−B, with Bnot necessarily effective; but, by Lemma 2.5 of [128], see also Lemma 3.39 of [257], and since −Bis h-nef, we conclude that Bis effective. We establish the claim by setting U:= B+h∗C. Observe that, under the isomorphism of homology groups, f∗(KZ+W)corre- sponds to (KZ+ˆ X)|ˆ X=Kˆ X, in particular we have Kn ˆ X=f∗(KZ+W)n=(KW+U)n. If we assume that KWis ample, then (KW+U)n≥(KW)n, equality holding if and only if U=0. Under assumption (**), it follows that Kn ˆ X=|G|Kn X=|G|Kn X=Kn W, hence U=0. Since however KWis ample, it follows that fis an embedding. 123
Topological methods in moduli theory 393 If instead we assume that KWhas the same class as f∗(KZ+W), we conclude first that necessarily B=0, and then we get that C=0. Hence Wis normal and has canonical singularities. Step 5 Finally, the group Gacts on W, preserving the direct summands of its fun- damental group . Hence, Gacts on the curve-factors, and on the locally symmetric factors. By assumption, moreover, it sends the summand ito itself, hence we get a well defined linear action on each Abelian variety A i, so that we have a diagonal linear action of Gon A. Since however the image of Wgenerates A, we can extend the action of Gon W to a compatible affine action on A. In the case where Gis abelian11, we can show that the real affine type of the action on Ais uniquely determined. This is taken care of by the following lemma. Lemma 150 Given a diagonal special Inoue type manifold, if G is abelian, the real affine type of the action of G on the Abelian variety A =(A1×···×Ar)is determined by the fundamental group exact sequence 1→=r i=1(i)×h j=1(πgj)×s l=1(π1(Ml)) →π1(X)→G→1. Proof Define as before := r i=1i=π1(A); moreover, since all the summands on the left hand side are normal in π1(X),set G:= π1(X)/((h j=1πgj)×(s l=1π1(Ml))). Observe now that Xis the quotient of its universal covering ˜ X=Cm× h j=1 Hj× s l=1 Dl by its fundamental group, acting diagonally (here Hjis a copy of Poincaré’s upper half plane, while Dlis an irreducible bounded symmetric domain of dimension at least two), hence we obtain that Gacts on Cmas a group of affine transformations. This action yields a homomorphism α:GIm(α) =: ˆ G⊂Aff(m,C): let Kthe kernel of α, and let G1:= ker(αL:G→GL(m,C)). G1is obviously Abelian, and contains , and maps onto a lattice ⊂ˆ G. 11 In [35,37] we made this assertion for any finite group G, see however Remark 23. 123
394 F. Catanese Since injects into ,∩K=0, whence Kinjects into G, therefore Kis a torsion subgroup; since is free, we obtain G1=⊕K, and we finally get K=Tors(G1), ˆ G=G/Tors(G1). Since our action is diagonal, we can write =⊕ r i=1 i, and the linear action of the group G2:= G/Kpreserves the summands. Since ˆ G⊂Aff(⊗R), we can now apply Proposition 21 to it. This shows that the affine group ˆ Gis uniquely determined. Finally, using the image groups G2,iof G2inside GL( i), we can define uniquely groups of affine transformations of Aiwhich fully determine the diagonal action of G on A(up to real affine automorphisms of each Ai). The proof of Theorem 148 is now completed. In order to obtain weak rigidity results, one has to use, as an invariant for group actions on tori and Abelian varieties, the Hodge type, introduced in Definition 19 (see also Remark 20). Remark 151 In the previous theorem special assumptions are needed in order to guar- antee that for each manifod Xhomotopy equivalent to Xthe classifying holomorphic map f:ˆ X→Zbe birational onto its image, and indeed an embedding. However, there is the possibility that an Inoue type variety deforms to one for which fis a covering of finite degree. This situation should be analyzed and the singularities of the image of fdescribed in detail, so as to lead to a generalization of the theory of Inoue type varieties, including ‘multiple’ Inoue type varieties (those for which fhas degree at least two). The study of moduli spaces of Inoue type varieties, and their connected and irre- ducible components, relies very much on the study of moduli spaces of varieties X endowed with the action of a finite group G: and it is for us a strong motivation to pursue this line of research. This topic will occupy a central role in the following sections, first in general, and then in the special case of algebraic curves. 9 Moduli spaces of surfaces and higher dimensional varieties Teichmüller theory works out quite fine in the case of projective curves, as well as other approaches, like Geometric Invariant Theory (see [178,179,302,304]), which provides a quasi-projective moduli space Mgendowed with a natural compactification Mg(this is a coarse moduli space for the so-called moduli stable projective curves: these are the reduced curves with at most nodes as singularities, such that their group of automorphisms is finite). 123
Topological methods in moduli theory 395 In higher dimensions one has a fully satisfactory theory of ‘local moduli’ for com- pact complex manifolds or spaces, but there are difficulties with the global theory. So, let us start from the local theory, developed by Kodaira and Spencer, and cul- minating in the results of Kuranishi and Grauert. 9.1 Kodaira–Spencer–Kuranishi theory While describing complex structures as integrable almost complex structures (see Theorem 32), it is convenient to view an almost complex structure as a differentiable (0,1)-form with values in the dual of the cotangent bundle (TY1,0)∨. This representation leads to the Kodaira–Spencer–Kuranishi theory of local defor- mations, addressing precisely the study of the small deformations of a complex manifold Y=(M,J0). In this theory, complex structures correspond to closed such forms which, by Dol- beault’s theorem, determine a cohomology class in H1(Y), where Yis the sheaf of holomorphic sections of the holomorphic tangent bundle TY1,0. We shall use here unambiguously the double notation TM 0,1=TY0,1,TM 1,0= TY1,0to refer to the splitting of the complexified tangent bundle determined by the complex structure J0. J0is a point in C(M), and a neighbourhood in the space of almost complex structures corresponds to a distribution of subspaces which are globally defined as graphs of an endomorphism φ:TM 0,1→TM 1,0, called a small variation of almost complex structure, since one then defines TM 0,1 φ:= {(u,φ(u))|u∈TM 0,1}⊂TM 0,1⊕TM 1,0. In terms of the old ¯ ∂operator, the new one is simply obtained by considering ¯ ∂φ:= ¯ ∂+φ, and the integrability condition is given by the Maurer–Cartan equation (MC)¯ ∂(φ) +1 2[φ,φ]=0. Observe that, since our original complex structure J0corresponds to φ=0, the derivative of the above equation at φ=0issimply ¯ ∂(φ) =0, hence the tangent space to the space of complex structures consists of the space of ¯ ∂-closed forms of type (0,1)and with values in the bundle TM 1,0. 123
396 F. Catanese One can restrict oneself (see e.g. [114]) to consider only the class of such forms φ in the Dolbeault cohomology group H1(Y):= ker(¯ ∂)/Im(¯ ∂), by looking at the action of the group of diffeomorphisms which are exponentials of global vector fields on M. Representing these cohomology classes by harmonic forms, the integrability equa- tion becomes easier to solve via the following Kuranishi equation. Let η1,...,η m∈H1(Y)be a basis for the space of harmonic (0,1)-forms with values in TM 1,0, and set t:= (t1,...,tm)∈Cm, so that t→ itiηiestablishes an isomorphism Cm∼ =H1(Y). Then the Kuranishi slice is obtained by associating to tthe unique power series solution of the following equation: φ(t)= i tiηi+1 2¯ ∂∗G[φ(t), φ (t)], satisfying moreover φ(t)=itiηi+higher order terms (Gdenotes here the Green operator). The upshot is that for these forms φ(t)the integrability equation simplifies drasti- cally; the result is summarized in the following definition. Definition 152 The Kuranishi space B(Y)is defined as the germ of complex subspace of H1(Y)defined by {t∈Cm|H[φ(t), φ (t)]=0}, where His the harmonic projector onto the space H2(Y)of harmonic forms of type (0,2)and with values in TM 1,0. Kuranishi space B(Y)parametrizes the set of small variations of almost complex structure φ(t)which are integrable. Hence over B(Y)we have a family of complex structures which deform the complex structure of Y. It follows then that the Kuranishi space B(Y)surjects onto the germ of the Teich- müller space at the point corresponding to the given complex structure Y=(M,J0). It fails badly in general to be a homeomorphism (see [86,114]): for instance Teich- müller space, in the case of the Hirzebruch Segre surface F2n(blow up of the cone onto the rational normal curve of degree 2n), consists of n+1 points p0,p2,..., p2n, such that the open sets are exactly the sets {p2i|i≤k}. But a first consequence is that Teichmüller space is locally connected by holomor- phic arcs, hence the determination of the connected components of C(M), respectively of T(M), can be done using the original definition of deformation equivalence, given by Kodaira and Spencer [248] (see Definition 109). One can define deformations not only for complex manifolds, but also for complex spaces. Definition 153 (1) A deformation of a compact complex space Xis a pair consisting of 123
Topological methods in moduli theory 397 (1.1) a flat proper morphism π:X→Tbetween connected complex spaces (i.e., π∗:OT,t→OX,xis a flat ring extension for each xwith π(x)=t) (1.2) an isomorphism ψ:X∼ =π−1(t0):= X0of Xwith a fibre X0of π. (2.1) A small deformation is the germ π:(X,X0)→(T,t0)of a deformation. (2.2) Given a deformation π:X→Tand a morphism f:T→Twith f(t 0)=t0,thepull-back f∗(X)is the fibre product X:= X×TTendowed with the projection onto the second factor T(then X∼ =X 0). (3.1) A small deformation π:X→Tis said to be versal or complete if every small deformation π:X→Tis obtained from it via pull back; it is said to be semi-universal if the differential of f:T→Tat t 0is uniquely determined, and universal if the morphism fis uniquely determined. (4) Two compact complex manifolds X,Yaresaidtobedirectly deformation equivalent if there are (4i) a deformation π:X→Tof Xwith Tirreducible and where all the fibres are smooth, and (4ii) an isomorphism ψ:Y∼ =π−1(t1)=: X1of Ywith a fibre X1of π. Remark 154 The technical assumption of flatness replaces, for families of spaces, the condition on πto be a submersion, necessary in order that the fibres be smooth manifolds. Let’s however come back to the case of complex manifolds, observing that in a small deformation of a compact complex manifold one can shrink the base Tand assume that all the fibres are smooth. We can now state the results of Kuranishi and Wavrik [259,260,371] in the language of deformation theory. Theorem 155 (Kuranishi) Let Y be a compact complex manifold: then (I) the Kuranishi family π:(Y,Y0)→(B(Y), 0)of Y is semiuniversal. (II) (B(Y), 0)is unique up to (non canonical) isomorphism, and is a germ of analytic subspace of the vector space H 1(Y, Y), inverse image of the origin under a local holomorphic map (called Kuranishi map) k:H1(Y, Y)→H2(Y, Y) whose differential vanishes at the origin. Moreover the quadratic term in the Taylor development of k is given by the bilinear map H 1(Y, Y)×H1(Y, Y)→H2(Y, Y), called Schouten bracket, which is the composition of cup product followed by Lie bracket of vector fields. (III) The Kuranishi family is a versal deformation of Ytfor t ∈B(Y). (IV) The Kuranishi family is universal if H 0(Y, Y)=0. (V) (Wavrik) The Kuranishi family is universal if B(Y)is reduced and h0(Yt, Yt) := dim H0(Yt, Yt)is constant for t ∈B(Y)in a suitable neighbourhood of 0. The Kodaira Spencer map, defined soon below, is to be thought as the derivative of a family of complex structures. 123
398 F. Catanese Definition 156 The Kodaira Spencer map of a family π:(Y,Y0)→(T,t0)of com- plex manifolds having a smooth base Tis defined as follows: consider the cotangent bundle sequence of the fibration 0→π∗(1 T)→1 Y→1 Y|T→0, and the direct image sequence of the dual sequence of bundles, 0→π∗(Y|T)→π∗(Y)→T→R1π∗(Y|T). Evaluation at the point t0yields a map ρof the tangent space to Tat t0into H1(Y0, Y0), which is the derivative of the variation of complex structure. The Kodaira Spencer map and the implicit functions theorem allow to determine the Kuranishi space and the Kuranishi family in many cases. Corollary 157 Let Y be a compact complex manifold and assume that we have a family π:(Y,Y0)→(T,t0)with smooth base T , such that Y ∼ =Y0, and such that the Kodaira Spencer map ρt0surjects onto H 1(Y, Y). Then the Kuranishi space B(Y)is smooth and there is a submanifold T ⊂Twhich maps isomorphically to B(Y); hence the Kuranishi family is the restriction of πto T. The key point is that, by versality of the Kuranishi family, there is a morphism f:T→B(Y)inducing πas a pull back, and ρis the derivative of f: then one uses the implicit functions theorem. This approach clearly works only if Yis unobstructed, which simply means that B(Y)is smooth. In general it is difficult to describe the obstruction map, and even calculating the quadratic term is nontrivial (see [216] for an interesting example). Even if it is difficult to calculate the obstruction map, Kuranishi theory gives a lower bound for the ‘number of moduli’ of Y, since it shows that B(Y)has dimension ≥h1(Y, Y)−h2(Y, Y). In the case of curves H2(Y, Y)=0, hence curves are unobstructed; in the case of a surface S dimB(S)≥h1(S)−h2(S)=−χ(S)+h0(S)=10χ(OS)−2K2 S+h0(S). The above is the Enriques inequality ([157], see also [155], and [126] for an improvement: observe that Max Noether postulated equality), proved by Kuranishi in all cases and also for non algebraic surfaces. 9.2 Kuranishi and Teichmüller Ideally, we would like to have that Teichmüller space, up to now only defined as a topological space, is indeed a complex space, locally isomorphic to Kuranishi space. In fact, we already remarked that there is a locally surjective continuous map of B(Y)to the germ T(M)Yof T(M)at the point corresponding to the complex structure 123
Topological methods in moduli theory 399 yielding Y. For curves this map is a local homeomorphism, and this fact provides a complex structure on Teichmüller space. Whether this holds in general is related to the following definition. Definition 158 A compact complex manifold Yis said to be rigidified if Aut(Y)∩ Diff0(Y)={Id Y}. A compact complex manifold Yis said to be cohomologically rigidified if Aut (Y)→Aut (H∗(Y,Z)) is injective, and rationally cohomologically rigidified if Aut (Y)→Aut (H∗(Y,Q)) is injective. Remark 159 We refer to [68] for the proof that surfaces of general type with q≥3are cohomologically rigidified, and for examples showing that there are rigidified surfaces of general type which are not cohomologically rigidified. In fact, it is clear that there is a universal tautological family of complex structures parametrized by C(M)(the closed subspace C(M)of AC(M)consisting of the set of complex structures on M), and with total space UC(M):= M×C(M), on which the group Diff+(M)naturally acts, in particular Diff0(M). The main observation is that Diff0(M)acts freely on C(M)if and only if for each complex structure Yon Mthe group of biholomorphisms Aut (Y)contains no nontrivial automorphism which is differentiably isotopic to the identity. Thus, the condition of being rigidified implies that the tautological family of complex structures descends to a universal family of complex structures on Teichmüller space: UT(M):= (M×C(M))/Diff0(M)→C(M))/Diff0(M)=T(M), on which the mapping class group acts. Fix now a complex structure yielding a compact complex manifold Y, and compare with the Kuranishi family Y→B(Y). Now, we already remarked that there is a locally surjective continuous map of B(Y) to the germ T(M)Yof T(M)at the point corresponding to the complex structure yielding Y. The following was observed in [114]. Remark 160 If (1) the Kuranishi family is universal at any point (2) B(Y)→T(M)Yis injective (it is then a local homeomorphism at every point) then Teichmüller space has a natural structure of complex space. In many cases (for instance, complex tori) Kuranishi and Teichmüller space coin- cide, in spite of the fact that the manifolds are not rigidified. For instance we showed in [114]: 123
400 F. Catanese Proposition 161 (1) The continuous map π:B(Y)→T(M)Yis a local homeo- morphism between Kuranishi space and Teichmüller space if there is an injective continuous map f :B(Y)→Z, where Z is Hausdorff, which factors through π. (2) Assume that Y is a compact Kähler manifold and that the local period map f is injective: then π:B(Y)→T(M)Yis a local homeomorphism. (3) In particular, this holds if Y is Kähler with trivial or torsion canonical divisor. Remark 162 (1) The condition of being rigidified implies the condition H0(Y)=0 (else there is a positive dimensional Lie group of biholomorphic self maps), and is obviously implied by the condition of being cohomologically rigidified. (2) By the cited Lefschetz’ lemma compact curves of genus g≥2 are rationally cohomologically rigidified, and it is an interesting question whether compact complex manifolds of general type are rigidified (see [67] and [68] for recent progresses). (3) One can dispense with many assumptions, at the cost of having a more complicated result. For instance, Meersseman shows in [284], explaining the situation via classical examples of [81], that one can take a quotient of the several Kuranishi families (using their semi-universality), obtaining, as analogues of Teichmüller spaces, respectively moduli spaces, an analytic Teichmüller groupoid, respectively a Riemann groupoid, both independent up to analytic Morita equivalence of the chosen countable disjoint union of Kuranishi spaces. These lead to some stacks called by the author ‘analytic Artin stack’ (see [11] and [164] as a reference on Artin stacks). 9.3 Varieties with singularities For higher dimensional varieties moduli theory works better if one considers varieties with moderate singularities, rather than smooth ones (see [256]), as we shall illustrate in the next sections for the case of algebraic surfaces. The Kuranishi theory does indeed extend perfectly to all compact complex spaces, and Kuranishi’s theorem was generalized by Grauert (see [186,187],seealso[334] for the algebraic analogue). Theorem 163 Grauert’s Kuranishi type theorem for complex spaces. Let X be a compact complex space: then (I) there is a semiuniversal deformation π:(X,X0)→(T,t0)of X , i.e., a deforma- tion such that every small deformation π:(X,X 0)→(T,t 0)is the pull-back of πfor an appropriate morphism f :(T,t 0)→(T,t0)whose differential at t 0 is uniquely determined. (II) (T,t0)is unique up to isomorphism, and is a germ of analytic subspace of the vector space T1of first order deformations. (T,t0)is the inverse image of the origin under a local holomorphic map (called Kuranishi map) k:T1→T2 123
Topological methods in moduli theory 401 to the finite dimensional vector space T2(called obstruction space), and whose differential vanishes at the origin (the point corresponding to the point t0). If X is reduced, or if the singularities of X are local complete intersection singu- larities, then T1=Ext1(1 X,OX). If the singularities of X are local complete intersection singularities, then T2= Ext2(1 X,OX). Indeed, the singularities which occur for the canonical models of varieties of gen- eral type are called canonical singularities and are somehow tractable (see [326]); nevertheless the study of their deformations may present highly nontrivial problems. For this reason, we restrict now ourselves to the case of complex dimension two, where these singularities are easier to describe. 10 Moduli spaces of surfaces of general type 10.1 Canonical models of surfaces of general type. The classification theory of algebraic varieties proposes to classify the birational equiv- alence classes of projective varieties. Now, in the birational class of a non ruled projective surface there is, by the the- orem of Castelnuovo (see e.g. [55]), a unique (up to isomorphism) minimal model S (concretely, minimal means that Scontains no (−1)-curves, i.e., curves Esuch that E∼ =P1, and with E2=−1). We shall assume from now on that Sis a smooth minimal (projective) surface of general type: this is equivalent (see [55]) to the two conditions: (*) K2 S>0 and KSis nef, where as well known, a divisor Dis said to be nef if, for each irreducible curve C,wehaveD·C≥0. It is very important that, as shown by Kodaira [250], the class of non minimal surfaces is stable by small deformation; on the other hand, a small deformation of a minimal algebraic surface of general type is again minimal (see prop. 5.5 of [21]). Therefore, the class of minimal algebraic surfaces of general type is stable by defor- mation in the large. Even if the canonical divisor KSis nef, it does not however need to be an ample divisor, indeed The canonical divisor K Sof a minimal surface of general type S is ample iff there does not exist an irreducible curve C on S with K ·C=0⇔there is no (−2)-curve ConS,i.e., a curve such that C ∼ =P1,and C2=−2. The number of (−2)-curves is bounded by the rank of the Neron Severi lattice NS(S)⊂H2(S,Z)/Torsion of S(NS(S)is the image of Pic(S)=H1(O∗ S) inside H2(S,Z)/Torsion, and, by Lefschetz’ (1,1) theorem, it is the intersection with the Hodge summand H1,1); the (−2)-curves can be contracted by a contraction π:S→X, where Xis a normal surface which is called the canonical model of S. The singularities of Xare called Rational Double Points (also called Du Val or Kleinian singularities, see [142]), and Xis a Gorenstein variety , i.e. (see [211]) the dualizing sheaf ωXis invertible, and the associated Cartier divisor KX, called again canonical divisor, is ample and such that π∗(KX)=KS. 123
402 F. Catanese Xis also the projective spectrum (set of homogeneous prime ideals) of the canonical ring R(S):= R(S,KS):= m≥0 H0(OS(mKS). This definition generalizes to any dimension, since for a variety of general type (one for which there is a pluricanonical map which is birational onto its image) the canonical ring is a finitely generated graded ring, as was proven by Birkar, Cascini, Hacon, and McKernan [43]. And one defines a canonical model as a variety with canonical singularities and with ample canonical divisor KX. More concretely, the canonical model of a surfaceof general type is directly obtained as the image, for m≥5, of the m-th pluricanonical map of S(associated to the sections in H0(OS(mKS))) as shown by Bombieri [54] (extended to any characteristic by Ekedahl [151], and in [93,94]). Theorem 164 (Bombieri) Let S be a minimal surface of general type, and consider the m-th pluricanonical map ϕmof S (associated to the linear system |mKS|)for m≥5,orform =4when K 2 S≥2. Then ϕmis a birational morphism whose image is isomorphic to the canonical model X, embedded by its m-th pluricanonical map. Corollary 165 Bombieri [55]Minimal surfaces S of general type with given topo- logical invariants e(S), b+(S)(here, b+(S)is the positivity of the intersetion form on H2(S,Z), and the pair of topological invariants is equivalent to the pair of holo- morphic invariants K 2 S,χ(S):= χ(OS)) are ‘bounded’, i.e., they belong to a finite number of families having a connected base. In particular, for fixed Euler number and positivity e(S), b+(S)we have a finite number of differentiable and topological types. In fact, one can deduce from here effective upper bounds for the number of these families, hence for the types (see [88,92]); see [273] for recent work on lower bounds and references to other recent and less recent results. Results in the style of Bombieri have been recently obtained by Hacon and Mck- ernan [205], and from these one obtains (non explicit) boundedness results for the varieties of general type with fixed invariants Kn X,χ(X). 10.2 The Gieseker moduli space When one deals with projective varieties or projective subschemes the most natural parametrization, from the point of view of deformation theory, is given by the Hilbert scheme, introduced by Grothendieck [200]. For instance, in the case of surfaces of general type with fixed invariants χ(S)=a and K2 S=b, their 5-canonical models X5are surfaces with Rational Double Points as singularities and of degree 25bin a fixed projective space PN, where N+1=P5:= h0(5KS)=χ(S)+10K2 S=a+10b. 123
Topological methods in moduli theory 403 The Hilbert polynomial of X5is the polynomial P(m):= h0(5mKS)=a+1 2(5m−1)5mb. Grothendieck [200] showed that, given a Hilbert polynomial (see [211]), there is (i) an integer dand (ii) a subscheme H=HPof the Grassmannian of codimension P(d)- subspaces of H0(PN,OPN(d)), called Hilbert scheme, such that (iii) Hparametrizes the degree dgraded pieces H0(I(d)) of the homogeneous ideals of all the subschemes ⊂PNhaving the given Hilbert polynomial P. The Hilbert point of is the Plücker point P(d)(r∨ )∈P(H0(PN,OPN(d))∨) where ris the restriction homomorphism (surjective for dlarge) r:H0(PN,OPN(d)) →H0(, O(d)). Inside Hone has the open sets H∗={|is smooth}⊂H0, where H0:= {|is reduced with only rational Gorenstein singularities}. Gieseker showed in [177], replacing the 5-canonical embeddding by an m-canonical embedding with much higher m, the following Theorem 166 (Gieseker) The moduli space of canonical models of surfaces of general type with invariants χ, K2exists as a quasi-projective scheme Mcan χ,K2 which is called the Gieseker moduli space. Similar theorems hold also for higher dimensional varieties, see [256,361], for the most recent developments, but we do not state here the results, which are more complicated and technical. 10.3 Components of moduli spaces and deformation equivalence We mentioned previously that the relation of deformation equivalence is a good general substitute for the condition that two manifolds belong to the same connected (resp.: irreducible) component of the moduli space. 123
404 F. Catanese Things get rather complicated and sometimes pathological in higher dimension, especially since, even for varieties of general type, there can be three models, smooth, terminal, and canonical model, and the latter are singular. In the case of surfaces of general type things still work out fine (up to a certain extent), since the main issue is to compare the deformations of minimal models versus the ones of canonical models. We have then the following theorem. Theorem 167 Given two minimal surfaces of general type S,Sand their respective canonical models X,X, then S and Sare deformation equivalent ⇔X and X are deformation equivalent ⇔ X and X yield two points in the same connected component of the Gieseker moduli space. One idea behind the proof is the observation that, in order to analyse deformation equivalence, one may restrict oneself to the case of families parametrized by a base Twith dim(T)=1: since two points in a complex space T⊂Cn(or in an algebraic variety) belong to the same irreducible component of Tif and only if they belong to an irreducible curve T⊂T. And one may further reduce to the case where Tis smooth simply by taking the normalization T0→Tred →Tof the reduction Tred of T, and taking the pull-back of the family to T0. A less trivial result which is used is the so-called simultaneous resolution of singu- larities (cf. [60–62,353]) Theorem 168 (Simultaneous resolution according to Brieskorn and Tjurina) Let T := Cτbe the basis of the semiuniversal deformation of a Rational Double Point (X,0). Then there exists a ramified Galois cover T →T , with T smooth T ∼ =Cτsuch that the pull-back X:= X×TTadmits a simultaneous resolution of singularities p : S→X(i.e., p is bimeromorphic, all the fibres of the composition S→X→T are smooth, and the fibre over t =0is equal to the minimal resolution of singularities of (X,0)). Another important observation is that the local analytic structure of the Gieseker moduli space is determined by the action of the group of automorphisms of Xon the Kuranishi space of X. Remark 169 Let Xbe the canonical model of a minimal surface of general type S with invariants χ,K2. The isomorphism class of Xdefines a point [X]∈Mcan χ,K2. Then the germ of complex space (Mcan χ,K2,[X])is analytically isomorphic to the quotient B(X)/Aut (X)of the Kuranishi space of Xby the finite group Aut (X)= Aut (S). Let Sbe a minimal surface of general type and let Xbe its canonical model. To avoid confusion between the corresponding Kuranishi spaces, denote by Def(S)the Kuranishi space for S, respectively Def(X)the Kuranishi space of X. Burns and Wahl [66], inspired by [13] explained the relation holding between Def(S)and Def(X). 123
Topological methods in moduli theory 405 Theorem 170 (Burns–Wahl) Assume that K Sis not ample and let π:S→Xbethe canonical morphism. Denote by LXthe space of local deformations of the singularities of X (Cartesian product of the corresponding Kuranishi spaces) and by LSthe space of deformations of a neighbourhood of the exceptional locus E xc(π ) of π. Then Def(S)is realized as the fibre product associated to the Cartesian diagram Def (S) //Def (SExc(π ))=: LS∼ =Cν, λ Def (X)//Def (XSing X )=: LX∼ =Cν, where νis the number of rational (−2)-curves in S, and λis a Galois covering with Galois group W := ⊕r i=1Wi, the direct sum of the Weyl groups Wiof the singular points of X (these are generated by reflections, hence yield a smooth quotient, see [124]). An immediate consequence is the following Corollary 171 (Burns–Wahl) (1) ψ:Def(S)→Def(X)is a finite morphism, in particular, ψis surjective. (2) If the derivative of Def(X)→LXis not surjective (i.e., the singularities of X cannot be independently smoothened by the first order infinitesimal deformations of X), then Def(S)is singular. In the next section we shall see the role played by automorphisms. 10.4 Automorphisms and canonical models Assume that Gis a group with a faithful action on a complex manifold Y: then Gacts naturally on the sheaves associated to 1, hence we have a linear representation of Gon the vector spaces that are the cohomology groups of such sheaves: for instance Gacts linearly on Hq(p Y), and also on the vector spaces H0((n Y)⊗m)=H0(OY(mKY)), hence on the canonical ring R(Y):= R(Y,KY):= m≥0 H0(OY(mKY)). If Yis a variety of general type, then Gacts linearly on the vector space H0(OY(mKY)), hence linearly on the m-th pluricanonical image Ym, which is an algebraic variety bimeromorphic to Y. Hence Gis isomorphic to a subgroup of the algebraic group Aut (Ym).Matsumura[283] used the structure theorem for linear alge- braic groups (see [218]) to show that, if Gwere infinite, then Aut(Ym)would contain a non trivial Cartan subgroup (Cor C∗), hence Ywould be uniruled: a contradiction. Hence we have the (already mentioned) theorem: Theorem 172 (Matsumura) The automorphism group of a variety Y of general type is finite. 123
406 F. Catanese The above considerations apply now to the m-th pluricanonical image Xof a variety Yof general type. I.e., we have an embedded variety X⊂P(V)and a linear representation of a finite group Gon the vector space V, such that Xis G-invariant (for Yof general type one has V:= Vm:= H0(OY(mKY))). Now, since we work over C, the vector space Vsplits uniquely, up to permutation of the summands, as a direct sum of irreducible representations (∗∗)Vm= ρ∈Irr(G) Wn(ρ) ρ. We come now to the basic notion of a family of G-automorphisms (this notion shall be further explained in the next section). Definition 173 A family of G-automorphisms is a triple (( p:X→T), G,α) where: (1) (p:X→T)is a flat family in a given category (a smooth family for the case of minimal models of surfaces of general type) (2) Gis a (finite) group (3) α:G×X→Xyields a biregular action G→Aut (X), which is compatible with the projection pand with the trivial action of Gon the base T(i.e., p(α(g,x)) = p(x), ∀g∈G,x∈X). As a shorthand notation, one may also write g(x)instead of α(g,x), and by abuse of notation say that the family of automorphisms is a deformation of the pair (Xt,G) instead of the triple (Xt,G,α t). The following then holds. Proposition 174 (1) A family of automorphisms of manifolds of general type (not necessarily minimal models) induces a family of automorphisms of canonical models. (2) A family of automorphisms of canonical models induces, if the basis T is con- nected, a constant decomposition type (∗∗)for Vm(t). (3) A family of automorphisms of smooth complex manifolds admits a differentiable trivialization, i.e., in a neighbourhood of t0∈T , a diffeomorphism as a family with (S0×T,pT,α 0×Id T); in other words, with the trivial family for which g(y,t)=(g(y), t). We refer to [114] for a sketchy proof, let us just observe that if we have a contin- uous family of finite dimensional representations, the multiplicity of an irreducible summand is a locally constant function on the parameter space T(being given by the scalar product of the respective trace functions, it is an integer valued continuous function, hence locally constant). 123
Topological methods in moduli theory 407 Let us then consider the case of a family of canonical models of varieties of general type: by 2) above, and shrinking the base in order to make the addendum R(p)m= p∗(OS(mK)) free, we get an embedding of the family (X,G)#→T×⎛ ⎝P⎛ ⎝Vm= ρ∈Irr(G) Wn(ρ) ρ⎞ ⎠,G⎞ ⎠. In other words, all the canonical models Xtare contained in a fixed projective space, where also the action of Gis fixed. Now, the canonical model Xtis left invariant by the action of Gif and only if its Hilbert point is fixed by G. Hence, we get a closed subset H0(χ , K2)Gof the Hilbert scheme H0(χ, K2) H0(χ, K2)G:= {X|ωX∼ =OX(1), Xisnormal,γ(X)=X∀γ∈G}. For instance in the case of surfaces one has the following theorem (see [256,360, 361] and references therein for results on moduli spaces of canonically polarized varieties in higher dimension). Theorem 175 The surfaces of general type which admit an action of a given pluri- canonical type (∗∗)i.e., with a fixed irreducible G- decomposition of their canonical ring, form a closed subvariety (Mcan χ,K2)G,(∗∗)of the moduli space Mcan χ,K2. Remark 176 The situation for the minimal models of surfaces of general type is dif- ferent, because then the subset of the moduli space where one has a fixed differentiable type is not closed, as showed in [33]. The puzzling phenomenon which we discovered in joint work with Ingrid Bauer [32, 33], on the moduli spaces of Burniat surfaces) is that deformations of automorphisms differ for canonical and for minimal models. More precisely, let Sbe a minimal surface of general type and let Xbe its canonical model. Denote by Def(S), resp. Def(X), the base of the Kuranishi family of S, resp. of X. Assume now that we have 1 = G≤Aut(S)=Aut(X). Then we can consider the Kuranishi space of G-invariant deformations of S, denoted by Def(S,G), and respectively consider Def(X,G); we have a natural map Def(S,G)→Def(X,G). We indeed show in [33] that this map needs not be surjective, even if surjectivity would seem plausible; we have the following result: Theorem 177 The deformations of nodal Burniat surfaces with K 2 S=4to extended Burniat surfaces with K 2 S=4yield examples where Def(S,(Z/2Z)2)→ Def(X,(Z/2Z)2)is not surjective. Moreover, whereas for the canonical model we have: Def(X,(Z/2Z)2)=Def(X), 123
408 F. Catanese for the minimal models we have Def(S,(Z/2Z)2)Def(S) and indeed the subset Def(S,(Z/2Z)2)corresponds to the locus of nodal Burniat surfaces. The moduli space of pairs (S,G)of an extended (or nodal)smooth projective and minimal Burniat surface S with K 2 S=4, taken together with its canonical G := (Z/2Z)2-action, is disconnected; but its image in the Gieseker moduli space is a connected open set. 10.5 Kuranishi subspaces for automorphisms of a fixed type Proposition 174 is quite useful when one analyses the deformations of a given G- action, say on a compact complex manifold: it tells us that we have to look at the complex structures for which the given differentiable action is holomorphic. Hence we derive (see [86]): Proposition 178 Consider a fixed action of a finite group G on a compact complex manifold X . Then we obtain a closed subset De f (X,G)of the Kuranishi space, cor- responding to deformations which preserve the given action, and yielding a maximal family of deformations of the G-action. The subset D e f (X,G)is the intersection De f (X)∩H1(X)G. Remark 179 (1) The proof is based on the well known and already cited Cartan’s lemma ([78]), that the action of a finite group in an analytic neighbourhood of a fixed point can be linearized. The proof and the result extend also to encompass compact complex spaces. (2) The above proposition is, as we shall now show in the example of projective curves, quite apt to estimate the dimension of the subspace Def (X,G)⊂De f (X).For instance, if De f (X)is smooth, then the dimension of De f (X,G)is just equal to dim(H1(X)G). We want to show how to calculate the tangent space to the Kuranishi space of G-invariant deformations in some cases. To do this, assume that Xis smooth, and that there is a normal crossing divisor D=∪ iDisuch that (1) Diis a smooth divisor (2) the Di’s intersect transversally (3) there is a cyclic sugbroup Gi⊂Gsuch that Di=Fix(Gi) (4) the stabilizer on the smooth locus of Di1∩Di2∩···∩ Dikequals Gi1⊕Gi2⊕ ···⊕Gik. Then, setting Y:= X/G,Yis smooth and the branch divisor Bis a normal crossing divisor B=∪ iBi. Define as usual the sheaf of logarithmic forms 1 X(log D1,...,log Dh),asthe sheaf generated by 1 X, and by the logarithmic derivatives of the equations siof the 123
Topological methods in moduli theory 409 divisors Di, define Y(−log D1,...,−log Dh)as the dual sheaf (this sheaf can also be more generally defined as the sheaf of derivations carrying the ideal sheaf of Dto itself). We have then the following proposition. Proposition 180 Let p :X→Y=X/G the quotient projection. Then p∗( X)= Y(−log B1,...,−log Bh). In particular, Hi(X)=Hi(Y(−log B1,...,−log Bh)). Idea of proof The basic idea of the proof is the calculation one does in dimension n=1 (and is the same one to be done at the generic point of the divisor Di, which maps to Bi). Namely, assume that the local quotient map is given by w=zm, the action being given by z→ z,being a primitive m-th root of unity. It follows that dz → dz, and, dually, (∂/∂z)→ −1(∂/∂ z). In particular, dlog(z)=dz/z=(1/m)dw/w and z(∂/∂ z)=mw(∂/∂w) are invariant. Then a vector field θ=f(z)∂/∂ z=f(z)/z(z∂/∂z)=f(z)/zmw(∂/∂w) is invariant if and only if f(z)/zis invariant, hence if and only if f(z)/z=F(w), where F(w) is holomorphic. For instance, in the case of algebraic curves, where De f (X)is smooth, the dimen- sion of De f (X,G)equals the dimension of H1(Y(−log B1,...,−log Bd)) =H1(Y(−y1,...,−yd)). Let gbe the genus of Y, and observe that this vector space is Serre dual to H0(2KY+ y1+ ···+ yd)). By Riemann-Roch its dimension equals 3g−3+dwhenever 3g−3+d≥0 (in fact H0(Y(−y1,...,−yd)) =0 when 2 −2g−d<0). Calculations with the above sheaves are the appropriate ones to calculate the defor- mations of ramified coverings, see for instance [32,82,95,317], and especially [33]. The main problem with the group of automorphisms on minimal models of surfaces (see [114]) is that a limit of isomorphisms need not be an isomorphism12, it can be for instance a Dehn twist on a vanishing cycle (see also [106,332]). 12 This means: there exists an algebraic family of surfaces S=∪ t∈TStover a smooth curve T,andan algebraic family of maps ϕt:St→St,t∈T, i.e., such that the union of the graphs of the ϕt’s is a closed algebraic set in the fibre product S×TS, with the property that ϕtis an isomorphism exactly for t= t0. 123
410 F. Catanese 11 Moduli spaces of symmetry marked varieties 11.1 Moduli marked varieties We give now the definition of a symmetry marked variety for projective varieties, but one can similarly give the same definition for complex or Kähler manifolds; to understand the concept of a marking, it suffices to consider a cyclic group acting on a variety X. A marking consists in this case of the choice of a generator for the group acting on X. The marking is very important when we have several actions of a group Gon some projective varieties, and we want to consider the diagonal action of Gon their product. Definition 181 (1) A G-marked (projective) variety is a triple (X,G,η) where Xis a projective variety, Gis a group and η:G→Aut(X)is an injective homomor- phism; (2) equivalently, a marked variety is a triple (X,G,α) where α:X×G→Xis a faithful action of the group Gon X. (3) Two marked varieties (X,G,α),(X,G,α )are said to be isomorphic if there is an isomorphism f:X→Xtransporting the action α:X×G→Xinto the action α:X×G→X, i.e., such that f◦α=α◦(f×id)⇔η=Ad(f)◦η, Ad(f)(φ) := fφf−1. (4) If Gis a subset of Aut(X), then the natural marked variety is the triple (X,G,i), where i:G→Aut(X)is the inclusion map, and it shall sometimes be denoted simply by the pair (X,G). (5) A marked curve (D,G,η)consisting of a smooth projective curve of genus gand a faithful action of the group Gon Dis said to be a marked triangle curve of genus gif D/G∼ =P1and the quotient morphism p:D→D/G∼ =P1is branched in three points. Remark 182 Observe that: (1) we have a natural action of Aut(G)on G-marked varieties, namely, if ψ∈ Aut (G), ψ(X,G,η) := (X,G,η ◦ψ−1). The corresponding equivalence class of a G-marked variety is defined to be a G-(unmarked)variety. (2) the action of the group Inn(G)of inner automorphisms does not change the isomor- phism class of (X,G,η) since, for γ∈G,wemayset f:= η(γ ),ψ:= Ad(γ ), and then η◦ψ=Ad(f)◦η, since η(ψ(g)) =η(γ gγ−1)=η(γ )η(g)(η(γ )−1)= Ad(f)(η(g)). (3) In the case where G=Aut(X), we see that Out(G)acts simply transitively on the isomorphism classes of the Aut(G)-orbit of (X,G,η). 123
Topological methods in moduli theory 411 In the spirit of the above concept we have already given, in the previous section, the definition of a family of G-automorphisms (we shall also speak of a family of G-marked varieties). The local deformation theory of a G-marked variety X, at least in the case where the group Gis finite, is simply given by the fixed locus Def(X)Gof the natural G -action on the Kuranishi space Def(X). As we mentioned previously (see 177), one encounters difficulties, when Xhas dimension at least 2, to use the Kuranishi approach for a global theory of moduli of G-marked minimal models. But we have a moduli space of G-marked varieties in the case of curves of genus g≥2 and in the case of canonical models of surfaces of general type, and similarly also for canonical models in higher dimension. In fact, one considers again a fixed linear representation space for the group G,the associated projective action, ⎛ ⎝P⎛ ⎝Vm= ρ∈Irr(G) Wn(ρ) ρ⎞ ⎠,G⎞ ⎠. and a locally closed subset of the Hilbert scheme Hcan (χ, K2)G⊂H0(χ, K2) Hcan (χ, K2)G:= {X|ωX∼ =OX(1), γ (X)=X∀γ∈G, Xisnormal,with canonical singularities }. In this case one divides the above subset by the subgroup C(G)⊂GL(Vm) which is the centralizer of G⊂GL(Vm). Remark 183 (1) Assume in fact that there is an isomorphism of the marked varieties (X,G)and (X,G)(here the marking is furnished by an m-th pluricanonical embed- ding of X,X#→P(Vm), and by the fixed action ρ:G#→GL(Vm)). Then there is an isomorphism f:X∼ =Xwith f◦γ=γ◦f∀γ∈G.Now, f induces a linear map of Vmwhich we denote by the same symbol: hence we can write the previous condition as f◦γ◦f−1=γ, ∀γ∈G⇔f∈C(G). (2) By Schur’s lemma C(G)=ρ∈Irr(G)GL(n(ρ), C)is a reductive group. (3) Observe that, since the dimension of Vmis determined by the holomorphic invariants of X, then there is only a finite number of possible representation types for the action of Gon Vm. (4) Assume moreover that X,Xare isomorphic varieties: then there is such a linear isomorphism f:Vm→Vmsending Xto X. 123
412 F. Catanese The variety Xhas therefore two G-markings, the marking η:G→Aut(X) provided by the pluricanonical embeddding X#→P(Vm), and another one given, in an evident notation, by Ad (f)◦η. Therefore Aut (X)contains two subgroups, G:= Im(η), and G:= Im(Ad(f)◦ η).G=Gif and only if flies in the normalizer NGof G. (5) This happens if Gis a full subgroup (this means that G=Aut(X)). In general fwill just belong to the normalizer NXof Aut (X), and NGshall just be the finite index subgroup of NXwhich stabilises, via conjugation, the subgroup G. (6) In other words, we define the marked moduli space (of a fixed representation type) as the quotient M[G]:=Hcan χ, K2G /CG. Then the forgetful map from the marked moduli space to the moduli space M[G]→ M, which factors through the quotient M(G)by NG/CG, yields finite maps M[G]→ M(G)→MG. We omit here technical details in higher dimension, which have been treated by Binru Li in his Bayreuth Ph.D. Thesis (in preparation). Observe that the action of Out(G)does not need to respect the representation type. Let us see now how the picture works in the case of curves: this case is already very enlightening and intriguing. 11.2 Moduli of curves with automorphisms There are several ‘moduli spaces’ of curves with automorphisms. First of all, given a finite group G, we define a subset Mg,Gof the moduli space Mgof smooth curves of genus g>1: Mg,Gis the locus of the curves that admit an effective action by the group G. It turns out that Mg,Gis a Zariski closed algebraic subset. In order to understand the irreducible components of Mg,Gwe have seen that Teichmüller theory plays an important role: it shows the connectedness, given an injective homomorphism ρ:G→Mapg, of the locus Tg,ρ := Fix(ρ (G)). Its image Mg,ρ in Mg,Gis a Zariski closed irreducible subset (as observed in [115]). Recall that to a curve Cof genus gwith an action by Gwe can associate several discrete invariants that are constant under deformation. The first is the above topological type of the G-action: it is a homomorphism ρ:G→Mapg, which is well-defined up to inner conjugation (induced by different choices of an isomorphism Map(C)∼ =Mapg). We immediately see that the locus Mg,ρ is first of all determined by the subgroup ρ(G)and not by the marking. Moreover, this locus remains the same not only if we change ρmodulo the action by Aut(G), but also if we change ρby the adjoint action by Mapg. 123
Topological methods in moduli theory 413 Definition 184 (1) The moduli space of G-marked curves of a certain topological type ρis the quotient of the Teichmüller submanifold Tg,ρ by the centralizer subgroup Cρ(G)of the subgroup ρ(G)of the mapping class group. We get a normal complex space which we shall denote Mg[ρ].Mg[ρ]=Tg,ρ /Cρ(G)is a finite covering of a Zariski closed subset of the usual moduli space (its image Mg,ρ ), therefore it is quasi-projective, by the theorem of Grauert and Remmert. (2) Defining Mg(ρ ) as the quotient of Tg,ρ by the normalizer Nρ(G)of ρ(G), we call it the moduli space of curves with a G-action of a given topological type. It is again a normal quasi-projective variety. Remark 185 (1) If we consider G:= ρ(G)as a subgroup G⊂Mapg, then we get a natural G-marking for any C∈Fix(G)=Tg,ρ . (2) As we said, Fix(G)=Tg,ρ is independent of the chosen marking, moreover the projection Fix(G)=Tg,ρ →Mg,ρ factors through a finite map Mg(ρ) → Mg,ρ . The next question is whether Mg(ρ) maps 1-1 into the moduli space of curves. This is not the case, as we shall easily see. Hence we give the following definition. Definition 186 Let G⊂Mapgbe a finite group, and let Crepresent a point in Fix(G). Then we have a natural inclusion G→AC:= Aut (C), and Cis a fixed point for the subgroup AC⊂Mapg:ACis indeed the stabilizer of the point Cin Mapg, so that locally (at the point of Mgcorresponding to C) we get a complex analytic isomorphism Mg=Tg/AC. We define HG:= ∩C∈Fix(G)ACand we shall say that Gis a full subgroup if G=HG. Equivalently, HGis the largest subgroup Hsuch that Fix(H)=Fix(G). This implies that HGis a full subgroup. Remark 187 The above definition shows that the map Tg,ρ →Mg,ρ factors through Mg(HG):= Tg,ρ /NHG, hence we restrict our attention only to full subgroups. Proposition 188 If H is a full subgroup H ⊂Mapg, and ρ:H⊂Mapgis the inclusion homomorphism, then Mg(ρ) is the normalization of Mg,ρ . Proof Recall that Mg(H)=Fix(H)/NH, and that for a general curve C∈Fix(H) we have AC=H. Assume then that C=γ(C), and that both C,C∈Fix(H). Then γACγ−1= AC, hence for Cgeneral γ∈NH. This means that the finite surjective morphism ϕ:Mg(H)→Mg,His generically bijective; Mg(H)is a normal variety, being locally the quotient of a smooth variety by a finite group: hence ϕis the normalization morphism and Mg(H)is the normalization of Mg,H. Next we investigate when the natural morphism Mg(H)→Mg,His not injective. In order that this be the case, we have already seen that we must pick up a curve C such that H⊂AC= H.Now,letCbe another point of Fix(H)which has the same image in the moduli space: this means that Cis in the orbit of Cso there is an element γ∈Mapgcarrying Cto C. 123
414 F. Catanese By conjugation γsends Hto another subgroup Hof AC. We can assume that H= H,elseγlies in the normalizer NHand we have the same point in Mg(H). Hence the points of Mg(H)that have the same image as Ccorrespond to the subset of subgroups H⊂ACwhich are conjugate to Hby the action of some γ∈Mapg. Example 189 Consider the genus 2 curve Cbirational to the affine curve with equa- tion y2=(x6−1). Its canonical ring is generated by x0,x1,yand is the quotient C[x0,x1,y]/(y2−x6 1+x6 0). Its group of automorphisms is generated (as a group of projective transforma- tions) by a(x0,x1,y):= (x0,x1,y)where is a primitive sixth root of 1, and by b(x0,x1,y):= (x1,x0,iy).ahas order 6, bhas order 4, the square b2is the hyperel- liptic involution h(x0,x1,y):= (x0,x1,−y). We have b◦a◦b−1=a−1, a formula which implies the (known) fact that the hyperelliptic involution lies in the centre of the group AC=Aut (C). Taking as Hthe cyclic subgroup generated by b, the space Fix(H)has dimension equal to 1 since C→C/H=P1is branched in 4 points. The quotient of AC= Aut (C)by b2is the dihedral group D6, and we see that His conjugate to 6 different subgroups of AC. Of course an important question in order to understand the locus in Mgof curves with automorphisms is the determination of all the non full subgroups G= HG. For instance Cornalba [127] answered this question for cyclic groups of prime order, thereby obtaining a full determination of the irreducible components of Sing(Mg). In fact, for g≥4, the locus Si ng(Mg)is the locus of curves admitting a nontrivial automorphism, so this locus is the locus of curves admitting a nontrivial automorphism of prime order. In this case, as we shall see in the next subsection, the topological type is easily determined. We omit to state Cornalba’s result in detail: it amounts in fact to a list of all non full such subgroups of prime order. We limit ourselves to indicate the simplest example of such a situation. Example 190 Consider the genus g=p−1 2curve C, birational to the affine curve with equation zp=(x2−1), where pis an odd prime number. Letting Hbe the cyclic group generated by the automorphisms (x,z)→ (x,z), where is a primitive p-th root of unity, we see that the quotient morphism C→C/H∼ =P1corresponds to the inclusion of fields C(x)#→C(C). The quotient map is branched on the three points x=1,−1,∞, so that Fi x(H)consists of just a point (the above curve C). An easy inspection of the above equation shows that the curve Cis hyperelliptic, with involution h:(x,z)→ (−x,z)which is hyperelliptic since C(C)h=C(z).In this case the locus Fi x(H)is contained in the hyperelliptic locus, and therefore it is not an irreducible component of Si ng(Mg). 123
Topological methods in moduli theory 415 11.3 Numerical and homological invariants of group actions on curves As already mentioned, given an effective action of a finite group Gon C,weset C:= C/G,g:= g(C), and we have the quotient morphism p:C→C/G=: C, aG-cover. The geometry of pencodes several numerical invariants that are constant on Mg,ρ (G): first of all the genus gof C, then the number dof branch points y1,...,yd∈C. We call the set B={y1,...,yd}the branch locus, and for each yiwe denote by mithe multiplicity of yi(the greatest number dividing the divisor p∗(yi)). We choose an ordering of Bsuch that m1≤···≤md. These numerical invariants g,d,m1≤ ··· ≤ mdform the so-called primary numerical type. p:C→Cis determined (Riemann’s existence theorem) by the monodromy, a surjective homomorphism: μ:π1(C\B)→G. We have: π1(C\B)∼ =g,d:= γ1,...,γ d,α 1,β 1,...,α g,β g| d i=1 γi g j=1[αj,βj]=1. We set then ci:= μ(γi),aj:= μ(α j),bj:= μ(β j), thus obtaining a Hurwitz generating vector, i.e. a vector v:= (c1,...,cd,a1,b1,...,ag,bg)∈Gd+2g s.t. •Gis generated by the entries c1,...,cd,a1,b1,...,ag,bg, •ci= 1G,∀i, and •d i=1cig j=1[aj,bj]=1. We see that the monodromy μis completely equivalent, once an isomorphism π1(C\B)∼ =g,dis chosen, to the datum of a Hurwitz generating vector (we also call the sequence c1,...,cd,a1,b1,...,ag,bgof the vector’s coordinates a Hurwitz generating system). A second numerical invariant of these components of Mg(G)is obtained from the monodromy μ:π1(C\{y1,..., yd})→Gof the restriction of pto p−1(C\{y1,..., yd}), and is called the ν-type or Nielsen function of the covering. The Nielsen function νis a function defined on the set of conjugacy classes in G which, for each conjugacy class Cin G, counts the number ν(C)of local monodromies c1,...,cdwhich belong to C(observe that the numbers m1≤···≤mdare just the orders of the local monodromies). 123
416 F. Catanese Observe in fact that the generators γjare well defined only up to conjugation in the group π1(C\{y1,..., yd}), hence the local monodromies are well defined only up to conjugation in the group G. We have already observed that the irreducible closed algebraic sets Mg,ρ (G)depend only upon what we call the ‘unmarked topological type’, which is defined as the conjugacy class of the subgroup ρ(G)inside Mapg. This concept remains however still mysterious, due to the complicated nature of the group Mapg. Therefore one tries to use more geometry to get a grasp on the topological type. The following is immediate by Riemann’s existence theorem and the irreducibil- ity of the moduli space Mg,dof d-pointed curves of genus g.Givengand d, the unmarked topological types whose primary numerical type is of the form g,d,m1,...,mdare in bijection with the quotient of the set of the corresponding monodromies μmodulo the actions by Aut (G)and by Map(g,d). Here Map(g,d)is the full mapping class group of genus gand dunordered points. Thus Riemann’s existence theorem shows that the components of the moduli space M(G):= ∪gMg(G) with numerical invariants g,dcorrespond to the following quotient set. Definition 191 A(g,d,G):= Epi(g,d,G)/Mapg,d×Aut(G). Thus a first step toward the general problem consists in finding a fine invariant that distinguishes these orbits. In the paper [115] we introduced a new homological invariant ˆfor G-actions on smooth curves (and showed that, in the case where Gis the dihedral group Dnof order 2n,ˆis a fine invariant since it distinguishes the different unmarked topological types). This invariant generalizes the classical homological invariant in the unramified case. Definition 192 Let p:C→C/G=: Cbe unramified, so that d=0 and we have a monodromy μ:π1(C)→G. Since Cis a classifying space for the group πg, we obtain a continuous map m:C→BG,π 1(m)=μ. Moreover, H2(C,Z)has a natural generator [C], the fundamental class of C determined by the orientation induced by the complex structure of C. The homological invariant of the G-marked action is then defined as: := H2(m)([C])∈H2(BG,Z)=H2(G,Z). If we forget the marking we have to take as an element in H2(G,Z)/Aut(G). 123
Topological methods in moduli theory 417 Proposition 193 Assume that a finite group G has a fixed point free action on a curve of genus g ≥3. Let p :C→C/G=: Cbe the quotient map and pick an isomorphism π1(C)∼ = πg. Let μ:πg→G be the surjection corresponding to the monodromy of p, and denote by ai:= μ(αi), bj:= μ(β j), i,j=1,...,g. Assume that we have a realization G =F/R of the group G as the quotient of a free group F, and that ai, bjare lifts of ai,bjto F. Then the homological invariant of the covering is the image of = g 1[ai, bi]∈[F,F]∩R into H2(G,Z)=([F,F]∩R)/[F,R]. Proof Cis obtained attaching a 2-cell Dto a bouquet of circles, with boundary ∂D mapping to g 1[αi,β i]. Similarly the 2-skeleton BG2of BG is obtained from a bouquet of circles, with fundamental group ∼ =F, attaching 2-cells according to the relations in R. Since g 1[ai,bi]=1inπg, the relation g 1[ai, bi]is a product of elements of R.Inthis way mis defined on D, and the image of the fundamental class of Cis the image of D, which is a sum of 2-cells whose boundary is exactly the loop . If we write as a sum of 2-cells, we get an element in H2(BG2,BG1,Z): but since is a product of commutators, the boundary of the corresponding 2-chain is indeed zero, so we get an element in H2(BG,Z)=H2(G,Z). 11.4 The refined homology invariant in the ramified case Assume now that p:C→C/Gis ramified. Then we define Hto be the minimal normal subgroup of Ggenerated by the local monodromies c1,...,cd⇔His the (normal) subgroup generated by the transformations γ∈Gwhichhavesomefixed point. We have therefore a factorization of p C→C := C/H→C:= C/G where C →Cis an unramified G/H-cover, and where C→C is totally ramified. The refined homology invariant includes and extends two invariants that have been studied in the literature, and were already mentioned: the ν-type (or Nielsen type) of the cover (also called shape in [170]) and the class in the second homology group H2(G/H,Z)corresponding to the unramified cover p:C =C/H→C. The construction of the invariant ˆis similar in spirit to the procedure used in the unramified case. But we achieve a little less in the ‘branched’ case of a non-free action. In this case we are only able to associate, to two given actions with the same ν-type, an invariant in a quotient group of H2(G,Z)which is the ‘difference’ of the respective ˆ-invariants. Here is the way we do it. 123
418 F. Catanese Let be the Riemann surface (with boundary) obtained from Cafter removing open discs iaround each of the branch points. Take generators γ1,...,γ dformed by a simple path going from the base point y0 to a point zion the circle bounding the open discs i, and by the circle ∂i. Fix then a CW-decomposition of as follows. The 0-skeleton 0consists of the point y0and of the points zi,i=1,...,d. The 1-skeleton 1is given by the geometric basis α1,...,β g,γ 1,...,γ dand the 2-skeleton 2consists of one cell. The restriction pof p:C→Cto p−1( ) is an unramified G-covering of and hence corresponds to a continuous map Bp:→BG, well defined up to homotopy. Let Bp1:→BG be a cellular approximation of Bp. Since Bp1can be regarded as a map of pairs Bp1:(, ∂ ) →(BG,BG1), the push-forward of the fundamental (orientation) class [, ∂]gives an element Bp1∗[, ∂]∈H2(BG,BG1)=R [F,R]. This element depends on the chosen cellular approximation Bp1of Bp, but its image in a quotient group which we denote by Kdoes not depend on the chosen cellular approximation, as shown in [116]. The main idea to define this quotient group is that a homotopy between two 1-cellular approximations can also be made cellular, hence the difference in relative homology is the sum of boundaries of cylinders; and if we have a cylinder Cwith upper circle a, lower circle c(with the same orientation), and meridian b, then the boundary of the corresponding 2-cell is ∂C=abc−1b−1. Definition 194 (1) For any finite group G,letFbe the free group generated by the elements of Gand let RFbe the subgroup of relations, that is G=F/R. Denote by ˆg∈Fthe generator corresponding to g∈G. For any ⊂G, union of non trivial conjugacy classes, let Gbe the quotient group of Fby the minimal normal subgroup Rgenerated by [F,R]and by the elements ˆaˆ bˆc−1ˆ b−1∈F, for any a,c∈,b∈G, such that b−1ab =c. (2) Define instead K=R/R⊂G=F/R, a central subgroup of G. We have thus a central extension 1→K→G→G→1. (3) Define H2,(G)=ker G→G×Gab . 123
Topological methods in moduli theory 419 Notice that H2(G,Z)∼ =R∩[F,F] [F,R]∼ =ker F [F,R]→G×Gab ∅. Remark 195 In particular, when =∅,H2,(G)∼ =H2(G,Z). By [115, Lemma 3.12] we have that the morphism R∩[F,F]→ R R ,r→ rR induces a surjective group homomorphism H2(G,Z)→H2,(G). Now,toagivenG-cover p:C→Cwe associate the set of the local mon- odromies, i.e., of the elements of Gwhich can be geometrically described as those which (i) stabilize some point xof Cand (ii) act on the tangent space at xby a rotation of angle 2π mwhere mis the order of the stabilizer at x. In terms of the notation that we have previously introduced =vis simply the union of the conjugacy classes of the ci’s. Definition 196 The tautological lift ˆvof vis the vector: c1,..., cd;a1, b1,...,ag, bg. Finally, define (v) as the class in Kof d 1cj· g 1ai, bi. It turns out that the image of (v) in K/Inn(G)is invariant under the action of Map(g,d),asshownin[115]. Moreover the ν-type of a Hurwitz monodromy vector vcan be deduced from (v), as it is essentially the image of (v) in the abelianized group Gab . Proposition 197 Let vbe a Hurwitz generating vector and let v⊂G be the union of the conjugacy classes of the c j’s , j ≤d . The abelianization G ab vof Gvcan be described as follows: Gab v∼ =⎛ ⎝ C⊂v ZC⎞ ⎠⊕⎛ ⎝ g∈G\v Zg⎞ ⎠, where Cdenotes a conjugacy class of G. Moreover the Nielsen function ν(v) coincides with the vector whose C-components are the corresponding components of the image of (v) ∈Gvin Gab v. 123
420 F. Catanese 11.5 Genus stabilization of components of moduli spaces of curves with G-symmetry In order to take into account also the automorphism group Aut (G), one has to consider K∪:= K, the disjoint union of all the K’s. Now, the group Aut(G)acts on K∪and we get a map ˆ:A(g,d,G)→(K∪)/ Aut (G) which is induced by v→ (v). Next, one has to observe that the Nielsen functions of coverings have to satisfy a necessary condition, consequence of the relation d i=1 ci g j=1[aj,bj]=1. Definition 198 An element ν=(nC)C∈ C={1} NC is admissible if the following equality holds in the Z-module Gab: C nC·[C]=0 (here [C]denotes the image element of Cin the abelianization Gab). The main result of [116] is the following ‘genus stabilization’ theorem. Theorem 199 There is an integer h such that for g>h ˆ:A(g,d,G)→(K∪)/ Aut (G) induces a bijection onto the set of admissible classes of refined homology invariants. In particular, if g>h , and we have two Hurwitz generating systems v1,v 2 having the same Nielsen function, they are equivalent if and only if the ‘difference’ ˆ(v1)ˆ(v2)−1∈H2,(G)is trivial. The above result extends a nice theorem of Livingston [272], Dunfield and Thurston [141] in the unramifed case, where also the statement is simpler. 123
Topological methods in moduli theory 421 Theorem 200 Fo r g >> 0 ˆ:A(g,0,G)→H2(G,Z)/Aut (G) is a bijection. Remark 201 Unfortunately the integer hin Theorem 199, which depends on the group G, is not explicit. A key concept used in the proof is the concept of genus stabilization of a covering, which we now briefly explain. Definition 202 Consider a group action of Gon a projective curve C, and let C→ C=C/Gthe quotient morphism, with monodromy μ:π1(C\B)→G (here Bis as usual the branch locus). Then the first genus stabilization of the differ- entiable covering is defined geometrically by simply adding a handle to the curve C, on which the covering is trivial. Algebraically, given the monodromy homomorphism μ:π1(C\B)∼ =g,d:= γ1,...,γ d,α 1,β 1,...,α g,β g| × d i=1 γi g j=1[αj,βj]=1→G, we simply extend μto μ1:g+1,d→Gsetting μ1(αg+1)=μ1(βg+1)=1G. In terms of Hurwitz vectors and Hurwitz generating systems, we replace the vector v:= (c1,...,cd,a1,b1,...,ag,bg)∈Gd+2g by v1:= (c1,...,cd,a1,b1,...,ag,bg,1,1)∈Gd+2g+2. The operation of first genus stabilization generates then an equivalence relation among monodromies (equivalently, Hurwitz generating systems), called stable equiv- alence. The most important step in the proof, the geometric understanding of the invariant ∈H2(G,Z)was obtained by Livingston [272]. 123
422 F. Catanese Theorem 203 Two monodromies μ1,μ2are stably equivalent if and only if they have the same invariant ∈H2(G,Z). A purely algebraic proof of Livingston’s theorem was given by Zimmermann [380], while a nice sketch of proof was given by Dunfield and Thurston [141]. Idea of proof Since one direction is clear (the map to BG is homotopically trivial on the handle that one adds to C), one has to show that two coverings are stably equivalent if their invariant is the same. The first idea is then to interpret second homology as bordism: given two maps of two curves C 1,C 2→BG, they have the same invariant in H2(G,Z)(image of the fundamental classes [C 1],[C 2]) if and only if there is a 3-manifold Wwith boundary ∂W=C 1−C 2, and a continuous map f:W→BG which extends the two maps defined on the boundary ∂W=C 1−C 2. Assume now that there is relative Morse function for (W,∂W), and that one first adds all the 1-handles (for the critical points of negativity 1) and then all the 2-handles. Assume that at level twe have a curve C t, to which we add a 1-handle. Then the genus of C tgrows by 1, and the monodromy is trivial on the meridian α(image of the generator of the fundamental group of the cylinder we are attaching). Pick now a simple loop βmeeting αtransversally in one point with intersection number 1: since the monodromy μis surjective, there is a simple loop γdisjoint from αand βwith μ(γ ) =μ(β ). Replacing βby β:= βγ−1, we obtain that the monodromy is trivial on the two simple loops αand β. A suitable neighbourhood of α∪βis then a handle that is added to C t, and with trivial monodromy. This shows that at each critical value of the Morse function we pass from one monodromy to a stably equivalent one (for the case of a 2-handle, repeat the same argument replacing the Morse function Fby its opposite −F). 11.6 Classification results for certain concrete groups The first result in this direction was obtained by Nielsen [313] who proved that ν determines ρif Gis cyclic (in fact in this case H2(G,Z)=0!). In the cyclic case the Nielsen function for G=Z/nis simply a function ν: (Z/n)\{0}→N, and admissibility here simply means that i i·ν(i)≡0(mod n). The class of νis just the equivalence class for the equivalence relation ν(i)∼νr(i), ∀r∈(Z/n)∗, where νr(i):= ν(ri),∀i∈(Z/n). From the refined Nielsen realization theorem of [96](37) it follows that the com- ponents of Mg(Z/n)are in bijection with the classes of Nielsen functions (see also [113] for an elementary proof). Example 204 For instance, in the case n=3, the components of Mg(Z/3)correspond to triples of integers g,a,b∈Nsuch that a≡b(mod 3), and g−1=3(g−1)+a+b (a:= ν(1), b:= ν(2)). 123
Topological methods in moduli theory 423 For g=0 we have, if we assume a≤b(we can do this by changing the generator of Z/3), b=a+3r,r∈N, and 2a+3r=g+2, thus a≡−(g−1)(mod 3), and 2a≤g+2. In the case where g=0, two such components Na( labelled by aas above) have images which do not intersect in Mgas soon as g≥5. Otherwise we would have two Z/3-quotient morphisms p1,p2:C→P1and a birational map C→C ⊂P1×P1. Then Cwould be the normalization of a curve of arithmetic genus 4, so g≤4. The genus gand the Nielsen class (which refine the primary numerical type), and the homological invariant h∈H2(G/H,Z)(here His again the subgroup generated by the local monodromies) determine the connected components of Mg(G)under some restrictions: for instance when Gis abelian or when Gacts freely and is the semi-direct product of two finite cyclic groups (as it follows by combining results from [96,113,145,146]). Theorem 205 (Edmonds) νand h ∈H2(G/H,Z)determine ρfor G abelian. More- over, if G is split-metacyclic and the action is free, then h determines ρ. However, in general, these invariants are not enough to distinguish unmarked topo- logical types, as one can see already for non-free Dn-actions (see [115]). Already for dihedral groups, one needs the refined homological invariant ˆ. Theorem 206 ([115]) For the dihedral group G =Dnthe connected components of the moduli space Mg(Dn)are in bijection, via the map ˆ, with the admissible classes of refined homology invariants. Remark 207 In this case, the classification is simple: two monodromies with the same Nielsen function differ by an element in H2, (Dn), which is a quotient of H2(Dn,Z). This last group is 0 if nis odd, and ∼ =Z/2forneven. More precisely: H2, (Dn)={0}if and only if •nis odd or •nis even and contains some reflection or •nis even and contains the non-trivial central element. In the remaining cases, H2,(Dn)=Z/2Z. The above result completes the classification of the unmarked topological types for G=Dn, begun in [112]; moreover this result entails the classification of the irreducible components of the loci Mg,Dn(see the appendix to [115]). It is an interesting question: for which groups Gdoes the refined homology invariant ˆdetermine the connected components of Mg(G)? In view of Edmonds’ result in the unramified case, it is reasonable to expect a positive answer for split metacyclic groups (work in progress by Sascha Weigl) or for some more general metacyclic or metaabelian groups. As mentioned in [141], p. 499, the group G=PSL(2,F13)shows that, for g=2, in the unramified case there are different components with trivial homology invariant ∈H2(G,Z): these topological types of coverings are therefore stably equivalent but not equivalent. 123
424 F. Catanese 11.7 Sing(Mg)II: loci of curves with automorphisms in Mg Several authors independently found restrictions in order that a finite subgroup Hof Mapgbe a not full subgroup (Singerman [343], Ries [327], and Magaard, Shaska, Shpectorov, and Völklein [275]) We refer to lemma 4.1 of [275] for the proof of the following result. Theorem 208 (MSSV) Suppose H ⊂G⊂Mapgand assume Z := Fix(H)= Fix(G)⊂Tg, with H a proper subgroup of G , and let C ∈Z . Then δ:= dim(Z)≤3. (I) if δ=3, then H has index 2in G, and C →C/G is covering of P1branched on six points, P1,...,P6, and with branching indices all equal to 2. Moreover the subgroup H corresponds to the unique genus two double cover of P1branched on the six points, P1,...,P6(by Galois theory, intermediate covers correspond to subgroups of G bijectively). (II) If δ=2, then H has index 2in G, and C →C/G is covering of P1branched on five points, P1,...,P5, and with branching indices 2,2,2,2,c5(where obviously c5≥2). Moreover the subgroup H corresponds to a genus one double cover of P1 branched on four of the points P1,...,P5which have branching index 2. (III) If δ=1, then there are three possibilities. (III-a) H has index 2in G, and C →C/G is covering of P1branched on four points P1,...,P4, with branching indices 2,2,2,2d4, where d4>1. Moreover the subgroup H corresponds to the unique genus one double cover of P1branched on the four points, P1,..., P4. (III-b) H has index 2in G , and C →C/G is covering of P1branched on four points, P1,...,P4, with branching indices 2,2,c3,c4, where 2≤c3≤c4>2. Moreover the subgroup H corresponds to a genus zero double cover of P1branched on two points whose branching index equals 2. (III-c) H is normal in G, G/H∼ =(Z/2)2, moreover C →C/G is covering of P1 branched on four points, P1,...,P4, with branching indices 2,2,2,c4, where c4>2. Moreover the subgroup H corresponds to the unique genus zero cover of P1with group (Z/2)2branched on the three points P1,P2,P3whose branching index equals 2. The main point in the theorem above is a calculation via the Hurwitz formula, showing that the normalizer of Hin Gis nontrivial; from this follows the classification of the singular locus of Mg, due to Cornalba, and which we do not reproduce here (see [127], and also [113] for a slightly different proof). 11.8 Stable curves and their automorphisms, Sing(Mg) The compactification of the moduli space of curves Mgis given by the moduli space Mgof stable curves of genus g(see [136,179,304]). Definition 209 A stable curve of genus g≥2 is a reduced and connected projective curve C, not necessarily irreducible, whose singularities are only nodes, such that 123
Topological methods in moduli theory 425 (1) the dualizing sheaf ωChas degree 2g−2 (2) its group of automorphisms Aut(C)is finite (equivalently, each smooth irreducible component D⊂Cof genus zero intersects C\Din at least 3 points). Again the Kuranishi family De f (C)of a stable curve is smooth of dimension 3g−3, and the locus of stable curves which admit a given G-action forms a smooth submanifold of De f (C). Hence again the singular locus of Mgcorresponds to loci of curves with automor- phisms which do not form a divisor. Example 210 We say that Chas an elliptic tail if we can write C=C1∪E, where E is a smooth curve of genus 1 intersecting C1in only one point P. In this case Chas always an automorphism γof order 2: γis the identity on C1, and, if we choose Pas the origin of the elliptic curve (E,P),γis simply multiplication by −1. An easy calculation shows that such curves with an elliptic tail form a divisor inside Mg. We refer to [113] (the second part) for the explicit determination of the loci of stable curves admitting an action by a cyclic group of prime order, especially those contained in the boundary ∂Mg. From this follows the description of the singular locus Sing(Mg), which we do not reproduce here. An interesting question is to describe similar loci also for other groups. As mentioned, there are irreducible loci which are contained in the boundary, while other ones are the closures of irreducible loci in Mg. But it is possible that some loci which are disjoint in Mghave closures that meet in Mg. We end this section with an explicit example, due to Antonio F. Costa, Milagros Izquierdo, and Hugo Parlier (cf. [129]), but we give a different proof. We refer to the notation of Example 204 in the following theorem. Theorem 211 The strata Naand N3+aof Mg(Z/3)fulfill: (i) Na∩N3+a=∅ (ii) Na∩N3+a=∅in Mg. Proof By the fact that inside Kuranishi space the locus of curves with a given action of a group Gis a smooth manifold, hence locally irreducible, one sees right away that these strata must intersect in a point corresponding to a curve with at least two different automorphisms of order 3. A natural choice is to take a stable curve with an action of (Z/3)2. The simplest choice is to take first the Fermat cubic F:= (x0,x1,x2)|x3 0+x3 1+x3 2=0 with the action of (u,v) ∈(Z/3)2 (x0,x1,x2)→ (x0,ux1,vx2). 123
426 F. Catanese We let then Cbe a (Z/3)-covering of the projective line, of genus g−3 and with ν(1)=a. Denote by ψthe covering automorphism and take a point p∈Cwhich is not a ramification point: then glue the three points ψv(p), v ∈Z/3,with the three points (1,0,−v). In this way we get a stable curve Dof genus g, and with an action of (Z/3)2. We have two actions of Z/3onD, the first induced by the action of (0,1)on F and of ψon C, and the second induced by the action of (1,1)on Fand ψon C(it is immediate to observe that the glueing is respected by the action). Now, for the first action of the generator ¯ 1∈Z/3onF, the fixed points are the points x2=0, with local coordinate t=x2 x0, such that t→ t, while for the second action the fixed points are the points x0=0, with local coordinate t=x0 x1, such that t→ 2t. Hence the same curve has two automorphisms of order 3, the first with ν(1)=a+3, the second with ν(1)=a. As easily shown (cf. [113]), since, for any of these two actions of Z/3, the nodes are not fixed, then there exists a smoothing for both actions of Z/3 on the curve D. This shows that the strata Naand Na+3have Din their closure. 11.9 Branch stabilization and relation with other approaches When g=0 our Gis related to the group Gdefined in [170] (Appendix), where the authors give a proof of a theorem by Conway and Parker. Roughly speaking the theorem says that: if the Schur multiplier M(G)(which is isomorphic to H2(G,Z)) is generated by commutators, then the ν-type is a fine stable invariant, when g=0. Theorem 212 (Conway–Parker, loc.cit.) In the case g=0,letG =F/R where F is free, and assume that H2(G,Z)∼ =[F,F]∩R [F,R]is generated by commutators. Then there is an integer N such that if the numerical function νtakes values ≥N , then there is only one equivalence class with the given numerical function ν. Michael Lönne, Fabio Perroni and myself have been able to extend this result in the following way: Theorem 213 Assume that g=0, or more generally that the union of nontrivial conjugacy classes of G generates G. Then there exists an integer N , depending on , such that if two Hurwitz generating systems v, w satisfy ν(v) ≥Nχ,ν(w) ≥Nχ, where χis the characteristic function of the set of conjugacy classes corresponding to , the Nielsen functions ν(v) and ν(w) are equivalent ⇔vis equivalent to wand they yield the same point in A(g,d,G). It is still an open question how to extend the previous result without the condition that generates G. Similar results have been obtained by Kulikov and Kharlamov [240], who use geometric arguments for the construction of semigroups similar to the ones constructed by Fried and Völklein, while a purely algebraic construction of a group similar to our group Kcan be found in work of Moravec on unramified Brauer groups ([297]). 123
Topological methods in moduli theory 427 11.10 Miller’s description of the second homology of a group and developments Clair Miller found ([286]) another nice description of the second homology group H2(G,Z), as follows. Definition 214 Let G,Gbe the free group on all pairs x,ywith x,y∈G. Then there is a natural surjection of G,Gonto the commutator subgroup [G,G] sending x,yto the commutator [x,y]. Denote as in [286]byZ(G)the kernel of this surjection (it might have been better to denote it by Z(G,G)), and then denote by B(G)(it might have been better to denote it by B(G,G)) the normal subgroup of Z(G)normally generated by the following elements (1) x,y (2) x,yy,x (3) z,xyyx,zxx,z, where yx:= xyx−1, (4) zx,yxx,[y,z]y,z. Theorem 215 (Miller) There is a canonical isomorphism Z (G)/B(G)∼ =H2(G,Z). Clearly we have then an exact sequence 0→H2(G,Z)∼ =Z(G)/B(G)→G,G→[G,G]→0 which corresponds to the previously seen 0→H2(G,Z)=(R∩[F,F])/[F,R]→[F,F]/[F,R]→[G,G]→0. The relations by Miller were later slightly modified by Moravec [297]inamore symmetric fashion as follows: (1) g,g∼1 (2) g1g2,h∼gg1 2,hg1g1,h (3) g,h1h2∼g,h1gh1,hh1 2. In the case where we consider also a finite union of conjugacy classes, Moravec defined the following group Definition 216 (Moravec) G∧G=G,G/B, where Bis defined by the previous relation (1), (2), (3) and by the further relation (4)g,k∼1,∀k∈. The definition by Moravec and the one given in [116] are related, as the following easy proposition shows. 123
428 F. Catanese Proposition 217 There is an exact sequence 0→H2,(G,Z)→G∧G→[G,G]→0. Another important source of interest for these quotient groups of H2(G,Z)comes from rationality questions. In general, the stable cohomology of a finite group G, or more generally of an algebraic group G, is obtained in the following way. Let Vbe a finite dimensional representation of G, and let Ube an open set of Vwhere Gacts freely: then U/Gis considered as an algebraic approximation to a classifying space for G, and one can take the limit over such representations (Vand Vbegin smaller than their direct sum) of the cohomology groups of U/G.Thesame can of course be done also for homology groups and Chow groups (see [355]). The groups that are thus obtained are important to study the problem of rationality of the quotients V/G, or of their stable rationality (Xis said to be stably rational if there is an integer nsuch that X×Pnis rational). We refer to [20,50,51,297] and the literature cited there for more details. 12 Connected components of moduli spaces and the action of the absolute Galois group Let Xbe a complex projective variety: let us quickly recall the notion of a conjugate variety. Remark 218 (1) φ∈Aut(C)acts on C[z0,...zn], by sending P(z)=n i=0aizi→ φ(P)(z):= n i=0φ(ai)zi. (2) Let Xbe as above a projective variety X⊂Pn C,X:= {z|fi(z)=0∀i}. The action of φextends coordinatewise to Pn C, and carries Xto another variety, denoted Xφ, and called the conjugate variety. Since fi(z)=0 implies φ( fi)(φ (z)) = 0, we see that Xφ={w|φ( fi)(w) =0∀i}. If φis complex conjugation, then it is clear that the variety Xφthat we obtain is diffeomorphic to X; but, in general, what happens when φis not continuous? Observe that, by the theorem of Steiniz, one has a surjection Aut (C)→Gal(¯ Q/Q), and by specialization the heart of the question concerns the action of Gal(¯ Q/Q)on varieties Xdefined over ¯ Q. For curves, since in general the dimensions of spaces of differential forms of a fixed degree and without poles are the same for Xφand X, we shall obtain a curve of the same genus, hence Xφand Xare diffeomorphic. 123
Topological methods in moduli theory 429 12.1 Galois conjugates of projective classifying spaces General questions of which the first is answered in the positive in most concrete cases, and the second is answered in the negative in many cases, as we shall see, are the following. Question 219 Assume that X is a projective K (π , 1), and assume φ∈Aut(C). (A) Is then the conjugate variety Xφstill a classifying space K (π ,1)? (B) Is then π1(Xφ)∼ =π∼ =π1(X)? Since φis never continuous, there would be no reason to expect a positive answer to both questions (A) and (B), except that Grothendieck showed [201], see also [202]. Theorem 220 Conjugate varieties X ,Xφhave isomorphic algebraic fundamental groups π1(X)alg ∼ =π1(Xφ)alg, where π1(X)alg is the profinite completion of the topological fundamental group π1(X). We recall once more that the profinite completion of a group Gis the inverse limit ˆ G=limKfG(G/K), of the factor groups G/K,Kbeing a normal subgroup of finite index in G; and since finite index subgroups of the fundamental group correspond to finite unramified (étale) covers, Grothendieck [201] defined in this way the algebraic fundamental group for varieties over other fields than the complex numbers, and also for more general schemes. The main point of the proof of the above theorem is that if we have f:Y→X which is étale, also the Galois conjugate fφ:Yφ→Xφis étale ( fφis just defined taking the Galois conjugate of the graph of f, a subvariety of Y×X). Since Galois conjugation gives an isomorphism of natural cohomology groups, which respects the cup product, as for instance the Dolbeault cohomology groups Hp(q X), we obtain interesting consequences in the direction of question A) above. Recall the following definition. Definition 221 Two varieties X,Yare said to be isogenous if there exist a third variety Z, and étale finite morphisms fX:Z→X,fY:Z→Y. Remark 222 It is obvious that if Xis isogenous to Y, then Xφis isogenous to Yφ. Theorem 223 (i) If X is an Abelian variety, or isogenous to an Abelian variety, the same holds for any Galois conjugate Xφ. (ii) If S is a Kodaira fibred surface, then any Galois conjugate Sφis also Kodaira fibred. (iii) If X is isogenous to a product of curves, the same holds for any Galois conjugate Xφ. 123
430 F. Catanese Proof (i) Xis an Abelian variety if and only it is a projective variety and there is a morphism X×X→X,(x,y)→ (x·y−1), which makes Xa group (see [303], it follows indeed that the group is commutative). This property holds for Xif and only if it holds for Xφ. (ii) The hypothesis is that there is f:S→Bsuch that all the fibres are smooth and not all isomorphic: obviously the same property holds, after Galois conjugation, for fφ:Sφ→Bφ. (iii) It suffices to show that the Galois conjugate of a product of curves is a product of curves. But since Xφ×Yφ=(X×Y)φand the Galois conjugate of a curve Cof genus gis again a curve of the same genus g, the statement follows. Proceeding with other projective K(π, 1)’s, the question becomes more subtle and we have to appeal to a famous theorem by Kazhdan on arithmetic varieties (see [117, 118,235,236,287,362]). Theorem 224 Assume that X is a projective manifold with K Xample, and that the universal covering ˜ X is a bounded symmetric domain. Let τ∈Aut(C)be an automorphism of C. Then the conjugate variety Xτhas universal covering ˜ Xτ∼ =˜ X. Simpler proofs follow from recent results obtained together with Antonio Di Scala, and based on the Aubin–Yau theorem and the results of Berger [39]. These results yield a precise characterization of varieties possessing a bounded symmetric domain as universal cover, and can be rather useful in view of the fact that our knowledge and classification of these fundamental groups is not so explicit. To state them in detail would require some space, hence we just mention the simplest result (see [117]). Theorem 225 Let X be a compact complex manifold of dimension n with KXample. Then the following two conditions (1) and (1’), resp. (2) and (2’) are equivalent: (1) X admits a slope zero tensor 0= ψ∈H0(Smn(1 X)(−mKX)), (for some positive integer m); (1’) X∼ =/ , where is a bounded symmetric domain of tube type and is a cocompact discrete subgroup of Aut() acting freely. (2) X admits a semi special tensor 0= φ∈H0(Sn(1 X)(−KX)⊗η), where η is a 2-torsion invertible sheaf, such that there is a point p ∈X for which the corresponding hypersurface Fp:= {φp=0}⊂P(TXp)is reduced. (2’) The universal cover of X is a polydisk. Moreover, in case (1), the degrees and the multiplicities of the irreducible factors of the polynomial ψpdetermine uniquely the universal covering X=. 12.2 Connected components of Gieseker’s moduli space For the sake of simplicity we shall describe in this and the next subsection the action of the absolute Galois group on the set of connected components of the moduli space of surfaces of general type. We first recall the situation concerning these components. 123
Topological methods in moduli theory 431 As we saw, all 5-canonical models of surfaces of general type with invariants K2,χoccur in a big family parametrized by an open set of the Hilbert scheme H0 parametrizing subschemes with Hilbert polynomial P(m)=χ+1 2(5m−1)5mK2, namely the open set H0(χ, K2):= |is reduced with only R.D.P.s as singularities . We shall however, for the sake of brevity, talk about connected components Nof the Gieseker moduli space Ma,beven if these do not really parametrize families of canonical models. We refer to [108] for a more ample discussion of the basic ideas which we are going to sketch here. Ma,bhas a finite number of connected components, and these parametrize the deformation classes of surfaces of general type with numerical invariants χ(S)= a,K2 S=b. By the classical theorem of Ehresmann [148], deformation equivalent varieties are diffeomorphic, and moreover, via a diffeomorphism carrying the canonical class to the canonical class. Hence, fixed the two numerical invariants χ(S)=a,K2 S=b, which are determined by the topology of S(indeed, by the Betti numbers bi(S)of Sand by b+:= positivity of the intersection form on H2(S,R)), we have a finite number of differentiable types. For some time the following question was open: whether two surfaces which are ori- entedly diffeomorphic would belong to the same connected component of the moduli space. I conjectured (in [232]) that the answer should be negative, on the basis of some families of simply connected surfaces of general type constructed in [82] and later investigated in [84], [104] and [111]: these were shown to be homeomorphic by the results of Freedman (see [166,167]), and it was then relatively easy to show then [85] that there were many connected components of the moduli space corresponding to homeomorphic but non diffeomorphic surfaces. It looked like the situation should be similar even if one would fix the diffeomorphism type. Friedman and Morgan instead made the ‘speculation’ that the answer to the DEF= DIFF question should be positive (1987) (see [168]), motivated by the new examples of homeomorphic but not diffeomorphic surfaces discovered by Donaldson (see [138] and [139] for a survey on this topic). The question was finally answered in the negative, and in every possible way ([26, 98,104,239,278]). Theorem 226 (Manetti ’98, Kharlamov–Kulikov 2001, C. 2001, C.-Wajnryb 2004, Bauer–C.-Grunewald 2005) The Friedman–Morgan speculation does not hold true and the DEF = DIFF ques- tion has a negative answer. In my joint work with Wajnryb [104] the DEF = DIFF question was shown to have a negative answer also for simply connected surfaces (indeed for some of the families of surfaces constructed in [82]). 123
432 F. Catanese I refer to [106] for a rather comprehensive treatment of the above questions (and to [3,17,100,107,111] for the symplectic point of view, [37,137] for the special case of geometric genus pg=0). 12.3 Arithmetic of moduli spaces and faithful actions of the absolute Galois group A basic remark is that all the schemes involved in the construction of the Gieseker moduli space are defined by equations involving only Z-coefficients, since the defining equation of the Hilbert scheme is a rank condition for a multiplication map (see for instance [188]), and similarly the condition ω⊗5 ∼ =O(1)is also closed (see [303]) and defined over Z. It follows that the absolute Galois group Gal(Q,Q)acts on the Gieseker moduli space Ma,b. In particular, it acts on the set of its irreducible components, and on the set of its connected components. After an incomplete initial attempt in [28] in joint work with Ingrid Bauer and Fritz Grunewald, we were able in [29]toshow: Theorem 227 The absolute Galois group Gal(¯ Q/Q)acts faithfully on the set of connected components of the Gieseker moduli space of surfaces of general type, M:= ∪x,y∈N,x,y≥1Mx,y. Another result in a similar direction had been obtained by Easton and Vakil [143] using abelian coverings of the plane branched on union of lines. Theorem 228 The absolute Galois group Gal(¯ Q/Q)acts faithfully on the set of irreducible components of the Gieseker moduli space of surfaces of general type, M:= ∪x,y∈N,x,y≥1Mx,y. The main ingredients for the proof of Theorem 227 are the following ones. (1) Define, for any complex number a∈C\{−2g,0,1,...,2g−1},Caas the hyper- elliptic curve of genus g≥3 which is the smooth complete model of the affine curve of equation w2=(z−a)(z+2g)2g−1 i=0(z−i). Consider then two complex numbers a,bsuch that a∈C\Q: then Ca∼ =Cbif and only if a=b. (2) If a∈¯ Q, then by Belyi’s theorem [38] there is a morphism fa:Ca→P1which is branched only on three points, 0,1,∞. (3) The normal closure Daof fayields a triangle curve, i.e., a curve Dawith the action of a finite group Gasuch that Da/Ga∼ =P1, and Da→P1is branched only on three points. 123
Topological methods in moduli theory 433 (4) Take surfaces isogenous to a product S=(Da×D)/Gawhere the action of Ga on Dis free. Denote by Nathe union of connected components parametrizing such surfaces. (5) Take all the possible twists of the Ga-action on Da×Dvia an automorphism ψ∈Aut (Ga)(i.e., given the action (x,y)→ (γ x,γy), consider all the actions of the form (x,y)→ (γ x,ψ(γ)y). One observes that, for each ψas above, we get more connected components in Na. (6) Find by an explicit calculation (using (4) and (5)) that the subgroup of Gal(¯ Q/Q) acting trivially on the set of connected components of the moduli space would be a normal and abelian subgroup. (7) Finally, this contradicts a known theorem (cf. [169]). Remark 229 An interesting remark of a referee is that the meaning of the above theorem could be further elaborated as a parallel of the Drinfend theory of Galois representation, built on the theorem of Belyi (see [140]). We leave this as a future task. 12.4 Change of fundamental group Jean Pierre Serre proved in the 60’s [336] the existence of a field automorphism φ∈Gal(¯ Q/Q), and a variety Xdefined over ¯ Qsuch that Xand the Galois conjugate variety Xφhave non isomorphic fundamental groups. In [29] this phenomenon is vastly generalized, thus answering question B) in the negative. Theorem 230 If σ∈Gal(¯ Q/Q)is not in the conjugacy class of complex conjugation, then there exists a surface isogenous to a product X such that X and the Galois conjugate surface Xσhave non-isomorphic fundamental groups. Since the argument for the above theorem is not constructive, let us observe that, in work in collaboration with Bauer and Grunewald [27,29], we discovered wide classes of explicit algebraic surfaces isogenous to a product for which the same phenomenon holds. By the strong rigidity of locally symmetric spaces X=D/ whose universal covering Dis an irreducible bounded symmetric domain of dimension ≥2, similar phenomena should also occur in this case. Remark 231 Further developments have been announced in [183] by Gonzaléz-Diez and Jaikin-Zapirain: for instance the faithfulness of the action of the absolute Galois group on the discrete set of the moduli space corresponding to Beauville surfaces, and the extension of Theorem 230 to all automorphisms σdifferent from complex conjugation. 123
434 F. Catanese 13 Stabilization results for the homology of moduli spaces of curves and Abelian varieties We have seen that the moduli space of curves is a rational K(π , 1), since it can be written as a quotient of the Teichmüller space Tgof a closed oriented real 2-manifold Mof genus g Mg=Tg/Mapg,Tg:= CS(M)/Diff0(M), As a corollary, and as we saw, the rational cohomology of the moduli space is calculated by group cohomology: H∗(Mg,Q)∼ =H∗(Mapg,Q). Harer showed, using the concept of genus stabilization that we have already intro- duced in Sect. 10, that these cohomology groups stabilize with g; indeed, stabilization furnishes an inclusion of a space which is homotopically equivalent to Mginside Mg+1(alternatively, one may say that Mapg→Mapg+1, by letting the operation be trivial on the added handle). Theorem 232 (Harer [208]) Let Maps g,rbe the mapping class group of an ori- entable surface F of genus g with r boundary components and s punctures. Then, for g ≥3k−1,H k(Maps g,r,Z)is independent of g and r as long as r >0,for g≥3k, Hk(Maps g,r,Q)is independent of g and r for every r, and for g ≥3k+1, Hk(Maps g,r,Z)is independent of g and r for every r. The ring structure of the cohomology of the “stable mapping class group” is described by a conjecture of Mumford [307], that has been proven by Madsen and Weiss [274]. Theorem 233 (Mumford’s conjecture) The stable cohomology of the moduli space of curves is a polynomial algebra H∗(Map∞,Q)=Q[K1,K2,...] where the class Kiis the direct image (pg)∗(Ki+1)of the (i+1)-th power of the relative canonical divisor of the universal family of curves. These results paralleled earlier results of Borel [58] and Charney and Lee [120]on the cohomology of arithmetic varieties, such as the moduli space of Abelian varieties and some partial compactifications of them. For instance, in the case of the moduli space of Abelian varieties we have the following theorem by Borel. Theorem 234 (Borel) The stable cohomology of the moduli space of Abelian varieties is a polynomial algebra H∗(A∞,Q)=Q[λ1,λ 3,λ 5...] where the class λiis the i-th Chern class of the universal bundle H1,0. 123
Topological methods in moduli theory 435 The theme of homology (cohomology-) stabilization is indeed a very general one, which has been recently revived through the work of several authors, also in other contexts ( see for instance [131,153,158,203]). It would take too long to dwell here on this topic, which would deserve a whole survey article devoted to it. 13.1 Epilogue There are many other interesting topics which are very tightly related to the main theme of this article. For instance, there is a relation between symmetric differentials and the fundamental group ([52,53,64,241]). Brunebarbe, Klingler and Totaro showed indeed in [64] that some ‘linearity’ property of the fundamental group entails the existence of nontrivial symmetric dif- ferentials. Theorem 235 Let X be a compact Kähler manifold, and let k be any field. Assume that there is a representation ρ:π1(X)→GL(r,k) with infinite image. Then the symmetric algebra of X is nontrivial, ⊕m≥0H0(Sm(1 X)) = C. Another very interesting topic is Gromov’s h-principle, for which we refer the reader to [152], and [74]. Perhaps not only the author is tired at this point. Acknowledgments Thanks to Efim Zelmanov for the honour of inviting me to write this article, and for his encouragement and support during its preparation. Thanks also to Maurizio Cornalba: some parts of this survey, which are directed to a rather general public, are influenced by an unpublished draft which I wrote after a plenary lecture which I gave at the XIV Congress of the Italian Mathematical Union (Catania 1991). Thanks to Ingrid Bauer, Michael Lönne and Fabio Perroni, for the pleasure I got from collaborating with them and for their invaluable contributions to our exciting joint research. A substantial part of the paper was written when I was visiting KIAS as research scholar: I am very grateful for the excellent atmosphere and mathematical environment I found at KIAS. Lectures held in Bayreuth, KAIST Daejeon and Centre Henri Lebesgue in Angers were also very useful. Support of the Forschergruppe 790 ‘Classification of algebraic surfaces and compact complex manifolds’ of the DFG, and of the ERC Advanced Grant No. 340258, ‘TADMICAMT’ is gratefully acknowledged. Finally, thanks to the students and postdocs who listened to my lectures in Bayreuth for their remarks, and to Ingrid Bauer, Binru Li, Sascha Weigl, and especially Wenfei Liu, for pointing out misprints and corrections to be made. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 123
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