DOCSLIB.ORG

A Torelli Theorem for Moduli Spaces of Principal Bundles Over a Curve Tome 62, No 1 (2012), P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Torelli Theorem for Moduli Spaces of Principal Bundles Over a Curve Tome 62, No 1 (2012), P R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Indranil BISWAS & Norbert HOFFMANN A Torelli theorem for moduli spaces of principal bundles over a curve Tome 62, no 1 (2012), p. 87-106. <http://aif.cedram.org/item?id=AIF_2012__62_1_87_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 1 (2012) 87-106 A TORELLI THEOREM FOR MODULI SPACES OF PRINCIPAL BUNDLES OVER A CURVE by Indranil BISWAS & Norbert HOFFMANN (*) 0 Abstract. — Let X and X be compact Riemann surfaces of genus > 3, and 0 d let G and G be nonabelian reductive complex groups. If one component MG(X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic d0 0 0 to another component MG0 (X ), then X is isomorphic to X . Résumé. — Soient X et X0 des surfaces de Riemann compactes de genre au moins 3, et G et G0 des groupes complexes réductifs non abéliens. Si une compo- d sante MG(X) de l’espace des modules de G–fibrés principaux semi-stables sur X d0 0 0 est isomorphe à une composante MG0 (X ), alors X est isomorphe à X . Introduction Let X be a compact connected Riemann surface of genus gX . The clas- sical Torelli theorem says that the isomorphism class of X is uniquely de- termined by the isomorphism class of the Jacobian Pic0(X), together with its canonical principal polarization Θ. Several authors have studied nonabelian analogues, replacing Jacobians by moduli spaces of vector bundles. Suppose gX > 3. Given a line bundle L on X, let Mn,L(X) denote the moduli space of semistable vector bundles E over X of rank n with fixed determinant Vn E =∼ L. Then the isomorphism Keywords: Principal bundle, moduli space, Torelli theorem. Math. classification: 14D20, 14C34. (*) We are very grateful to the referee for pointing out a gap in an earlier version of this paper, and also for some valuable suggestions. We thank D. Prasad for a useful conversation. The first author wishes to thank the Freie Universität Berlin and the Institute for Mathematical Sciences at Chennai for hospitality. The second author was supported by the SFB 647: Raum - Zeit - Materie. He thanks TIFR (Mumbai) and IMSc (Chennai) for hospitality while some part of this work was done. 88 Indranil BISWAS & Norbert HOFFMANN class of X is uniquely determined by the isomorphism class of the projective variety Mn,L(X). This was first proved for n = 2 and deg(L) odd, by Mumford-Newstead [10, p. 1201, Corollary] and Tyurin [19, Theorem 1]. It was then proved more generally for n coprime to deg(L), by Tyurin [20, Theorem 1] and Narasimhan-Ramanan [12, Theorem 3]. Finally, this statement was proved in full generality by Kouvidakis- Pantev [6, Theorem E], and then later for gX > 4 also by Hwang-Ramanan [5, Theorem 5.1] and by Sun [18, Corollary 1.3]. These proofs used either an intermediate Jacobian of the moduli space [10, 12], or Higgs bundles and the Hitchin map [6,5], or rational curves on the moduli spaces [19, 20, 18]. Our aim here is to address a similar question for the moduli spaces of principal bundles. Given a connected reductive complex linear algebraic d group G, let MG(X) denote the moduli space of semistable principal G– bundles over X of topological type d ∈ π1(G). Theorem 0.1. — Let G and G0 be nonabelian connected reductive com- 0 0 0 plex groups. Let d ∈ π1(G) and d ∈ π1(G ) be given. Let X and X be com- d d0 0 pact Riemann surfaces of genus > 3. If MG(X) is isomorphic to MG0 (X ), then X is isomorphic to X0. For the proof, our strategy is fundamentally different from the earlier ones. Whereas [6] and [5] start from the stable locus, our starting point is the strictly semistable locus. We show that it lies in the singular locus and is characterized by its type of singularities. Using powers of its anticanonical line bundle, we map it to a projective space. The fibers of this map allow us to reconstruct the Jacobian of X and its principal polarization. Then the usual Torelli theorem applies. This method does not work if the strictly semistable locus is empty. But this case is rare, and can in fact be reduced to the earlier results on moduli spaces of vector bundles in the coprime case. The structure of this paper is as follows. Section1 deals with quotient singularities arising from a group action. We give a criterion to distinguish the case of a finite group from the torus C∗. d In Section2, we collect some basic facts about the moduli spaces MG(X). We use several tools valid only in characteristic 0, like representations of π1(X), and the stability of induced bundles. d Section3 describes the strictly semistable locus in MG(X). We relate it to moduli spaces of principal H–bundles for Levi subgroups H of maximal ANNALES DE L’INSTITUT FOURIER TORELLI FOR MODULI OF PRINCIPAL BUNDLES 89 parabolic subgroups in G. Furthermore, we characterize it by the singulari- d ties it contains. As a byproduct, we show that the smooth locus of MG(X) coincides with the regularly stable locus. In the final Section4, we prove Theorem 0.1, by reconstructing the com- d pact Riemann surface X from the projective variety MG(X). In fact we prove a slightly stronger result, namely that X is determined up to isomor- d phy by the smooth locus of MG(X); see Theorem 4.1. 1. Some quotient singularities Let S and S0 be schemes of finite type over C. We say that S is near 0 a closed point s0 ∈ S analytically isomorphic to S near a closed point 0 0 0 0 s0 ∈ S0 if for some open neighborhoods s0 ∈ U ⊆ S and s0 ∈ U ⊆ S in the Euclidean topology, there exists a biholomorphic isomorphism U → U 0 0 that maps s0 to s0. We always denote by Ssing ⊆ S the singular locus of S. Let G be a reductive linear algebraic group over C. We assume that G acts linearly on a finite-dimensional complex vector space V . Let p : V −→ S := V//G = Spec(Sym(V ∗)G) be the GIT–quotient, and put s0 := p(0) ∈ S. Lemma 1.1. — Let G be finite and nontrivial. Suppose that the fixed g locus V ⊆ V has codimension > 2 for all g ∈ G with g 6= 1. i) The quotient S = V//G is singular in s0 = p(0). ii) If s0 ∈ U ⊆ S is an open neighborhood in the Euclidean topology sing sing such that U \ S is connected, then π1(U \ S ) is nontrivial. Proof. — Using the assumption on codim V g,[11, Lemma 4.4] implies [ p−1(Ssing) = V g. g6=1 This proves (i). In the situation of (ii), p−1(U \ Ssing) is connected because −1 sing s0 has only one inverse image. Hence p (U \ S ) is a nontrivial covering sing of U \ S , which is therefore not simply connected. Lemma 1.2. — Let G = C∗ act linearly on the finite-dimensional com- plex vector space V , with associated weight space decomposition M V = Vm. m∈Z Suppose that V−1 and V1 both have dimension > 2. TOME 62 (2012), FASCICULE 1 90 Indranil BISWAS & Norbert HOFFMANN ∗ i) The quotient S = V//C is singular in s0 = p(0). ii) Every neighborhood s0 ∈ U ⊆ S in the Euclidean topology contains 0 0 sing an open neighborhood s0 ∈ U ⊆ U such that U \S is connected 0 sing and π1(U \ S ) is trivial. ∗ Proof. — It is easy to verify S = (V+ ⊕ V−)//C × V0 with M M V+ := Vm and V− := Vm. m>0 m<0 Replacing V by V+ ⊕ V− if necessary, we may thus assume V0 = 0 without loss of generality. Then the GIT–stable locus V stab ⊆ V is stab V = V \ (V+ ∪ V−), stab and the image p(V+ ∪ V−) of V \ V is just the point s0 ∈ S. Let µn M V = Vm ⊆ V m∈nZ ∗ be the fixed locus of the finite subgroup µn ⊆ C . We claim −1 sing [ µn (1.1) p (S ) = V+ ∪ V− ∪ V . n>2 To check this, we first consider a stable point v ∈ V stab. Its image p(v) ∈ S is smooth if and only if its stabilizer in C∗ is trivial, according to Luna’s étale slice theorem [8] and Lemma 1.1. µ stab Suppose next that V n ∩ V is non-empty for some n > 2. Then it is dense in V µn , but also contained in the closed subset p−1(Ssing). Hence µn −1 sing sing −1 sing 0 ∈ V ⊆ p (S ), which implies s0 ∈ S . It follows that p (S ) contains the unstable locus V+ ∪ V− in this case.
Load more

Recommended publications
  • Arxiv:2008.05229V2 [Math.AG]
    A TORELLI THEOREM FOR MODULI SPACES OF PARABOLIC VECTOR BUNDLES OVER AN ELLIPTIC CURVE THIAGO FASSARELLA AND LUANA JUSTO Abstract. Let C be an elliptic curve, w ∈ C, and let S ⊂ C be a finite subset of cardinality at least 3. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle OC (w) over (C,S) which are semistable with respect to a weight vector 1 1 2 ,..., 2 . 1. Introduction Let C be a smooth complex curve of genus g ≥ 2 and fix w ∈ C. Let M be the corresponding moduli space of semistable rank two vector bundles having OC (w) as determinant line bundle. A classical Torelli type theorem of D. Mumford and P. Newstead [MN68] says that the isomorphism class of M determines the isomorphism class of C. This result has been extended first in [Tyu70, NR75, KP95] to higher rank and later to the parabolic context, which we now describe. We now assume g ≥ 0. Let S ⊂ C be a finite subset of cardinality n ≥ 1. Let Ma be the moduli space of rank two parabolic vector bundles on (C,S) with fixed determinant line bundle OC (w), and which are µa-semistable, see Section 2. The subscript a refers to a particular choice of a weight vector a = (a1, . , an) of real numbers, 0 ≤ ai ≤ 1, which gives the slope-stability condition. The moduli 1 1 space associated to the central weight aF = 2 ,..., 2 is particularly interesting, for instance when g = 0 and n ≥ 5 it is a Fano variety that is smooth if n is odd and has isolated singularities if n is even, see [Muk05, Cas15, AM16, AFKM19].
    [Show full text]
  • Weighted Fano Varieties and Infinitesimal Torelli Problem 3
    WEIGHTED FANO VARIETIES AND INFINITESIMAL TORELLI PROBLEM ENRICO FATIGHENTI, LUCA RIZZI AND FRANCESCO ZUCCONI Abstract. We solve the infinitesimal Torelli problem for 3-dimensional quasi-smooth Q-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length 3 Gushel- Mukai chain of prime smooth Fanos of coindex 3 and degree 10. 1. Introduction Minimal model program suggests us to formulate inside the category of varieties with terminal singularities many questions which were initially asked for smooth varieties. The construction of the period map, and the related Torelli type problems are definitely among these problems. 1.1. Infinitesimal Torelli. A Q-Fano variety is a projective variety X such that X has at worst Q-factorial terminal singularities, −KX is ample and Pic(X) has rank 1; cf. [CPR00]. In this paper we give a full answer to the infinitesimal Torelli problem in the case of quasi- smooth Q-Fano hypersurfaces of dimension 3 with terminal singularities and with Picard number 1. In subsections 2.1 of this paper the reader can find a basic dictionary and up to date references needed to understand the statement of the infinitesimal Torelli problem and where to find the meaning of infinitesimal variation of Hodge structures (IVHS; to short). Here we stress only that the method à la Griffiths based on the extension of the Macaulay’s theorem in algebra to the weighted case, cf.
    [Show full text]
  • Algebraic Curves and Surfaces
    Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces .
    [Show full text]
  • Abelian Varieties
    Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties .
    [Show full text]
  • The Derived Torelli Theorem
    The derived Torelli theorem Pieter Belmans Abstract These are the notes for my lecture for the 2012–2013 course Surfaces K3 et théorie des déformations at Université Paris-Sud 11, and provide some back- ground to the actual lecture. The goal is to discuss Fourier-Mukai transforms, and apply the obtained techniques to K3 surfaces. The main result that will be discussed is a version of the global Torelli theorem in the context of derived categories, the so called derived Torelli theorem, which is theorem 40. The main reference for these notes (and everything else that is related to Fourier-Mukai transforms) is the excellent book [Huy06]. It is followed closely, to add a bit of originality some of the calculations left as an exercise are done explicitly. Contents 1 Derived categories in algebraic geometry2 1.1 Derived categories...............................2 1.2 Total derived functors.............................3 b 1.3 Relationship between D (X ) and the geometry of X ...........5 2 Fourier-Mukai transforms6 2.1 Definition.....................................6 2.2 Some examples.................................8 2.3 Properties of Fourier-Mukai transforms...................9 2.4 Bondal-Orlov theorem............................. 10 3 The derived Torelli theorem 11 3.1 Preliminaries: the K-theoretic Fourier-Mukai transform......... 11 3.2 Preliminaries: the cohomological Fourier-Mukai transform....... 12 3.3 Preliminaries: the Hodge structure..................... 14 3.4 Derived Torelli.................................. 15 3.5 Auto-equivalences of K3 surfaces...................... 15 1 1 Derived categories in algebraic geometry The subject of these notes is the notion of derived equivalence (see definition1), applied to the case of K3 surfaces (see §3).
    [Show full text]
  • Moduli of K3 Surfaces and Irreducible Symplectic Manifolds
    Moduli of K3 Surfaces and Irreducible Symplectic Manifolds V. Gritsenko, K. Hulek and G.K. Sankaran Abstract The name “K3 surfaces” was coined by A. Weil in 1957 when he formu- lated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been intro- duced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil’s programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. Contents 1 Introduction 2 2 Periods of K3 surfaces and the Torelli theorem 4 3 Irreducible symplectic manifolds 15 4 Projective models 24 arXiv:1012.4155v2 [math.AG] 26 May 2011 5 Compactifications 26 6 Modular forms and Kodaira dimension 39 7 Orthogonal groups and reflections 46 8 The quasi pull-back of modular forms 51 9 Arithmetic of root systems 59 1 1 Introduction The history of K3 surfaces goes back to classical algebraic geometry. In modern complex algebraic geometry a K3 surface is a compact complex surface S whose canonical bundle is trivial, i.e.
    [Show full text]
  • Algebraic Boundaries of Hilbert's SOS Cones
    COMPOSITIO MATHEMATICA Algebraic boundaries of Hilbert's SOS cones Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad and Bernd Sturmfels Compositio Math. 148 (2012), 1717{1735. doi:10.1112/S0010437X12000437 FOUNDATION COMPOSITIO MATHEMATICA Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 29 Sep 2021 at 03:34:16, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X12000437 Compositio Math. 148 (2012) 1717{1735 doi:10.1112/S0010437X12000437 Algebraic boundaries of Hilbert's SOS cones Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad and Bernd Sturmfels Abstract We study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether{Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids. 1. Introduction A fundamental object in convex algebraic geometry is the cone Σn;2d of homogeneous polynomials of degree 2d in R[x1; : : : ; xn] that are sums of squares (SOS). Hilbert [Hil88] showed that the cones Σ3;6 and Σ4;4 are strictly contained in the corresponding cones P3;6 and P4;4 of non-negative polynomials.
    [Show full text]
  • The Moduli of Curves of Genus Six and K3 Surfaces
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 3, March 2011, Pages 1445–1462 S 0002-9947(2010)05126-8 Article electronically published on October 25, 2010 THE MODULI OF CURVES OF GENUS SIX AND K3 SURFACES MICHELA ARTEBANI AND SHIGEYUKI KONDO¯ Abstract. We prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces. Introduction This paper gives a birational period map between the coarse moduli space M6 of curves of genus six and a bounded symmetric domain of type IV, parametrizing some lattice-polarized K3 surfaces. A similar correspondence, between a moduli space of curves and a period space of K3 surfaces, was given by the second author for curves of genus three and genus four in [Ko2] and [Ko3]. Some of the results in this paper were announced in [Ko3]. Realizing a moduli space as an arithmetic quotient of a bounded symmetric domain is interesting for several reasons. The first reason is that this gives a connection with the theory of automorphic forms. Recently Borcherds [Bor] gave a systematic method to construct automor- phic forms on a bounded symmetric domain of type IV. Allcock and Freitag [AF] and the second author [Ko4], [Ko5] used his theory to obtain explicit embeddings in a projective space for the moduli spaces of cubic surfaces, Enriques surfaces and curves of genus three, respectively. Secondly, an arithmetic quotient of a bounded symmetric domain has several compactifications, that is, Satake-Baily-Borel’s, Mumford’s toroidal and Looijenga’s compactifications [Lo1], [Lo2].
    [Show full text]
  • On a Proof of Torelli's Theorem
    Contents 0.1 Assumed Knowledge . 5 0.2 Outline . 5 Chapter 1 Sheaves and sheaf cohomology 7 1.1 Sheaves . 7 1.2 Sheaves associated with functions . 12 1.3 Sheaf cohomology . 14 1.4 Vanishing theorems . 22 n n 1.5 Cohomology of C and P ........................... 24 Chapter 2 Riemann surfaces 27 2.1 Properties of Riemann surfaces . 28 2.2 Examples of Riemann surfaces . 29 2.3 Cohomology of Riemann surfaces . 31 2.4 The Riemann-Hurwitz formula . 32 2.5 Hyperellipticity . 33 Chapter 3 The classical theorems of Abel and Jacobi 35 3.1 Divisors . 35 3.2 The Abel-Jacobi map and the Jacobian variety . 38 3.3 Line bundles . 41 3.4 Pic(S)...................................... 45 Chapter 4 Linear systems and the Riemann-Roch theorem 51 4.1 The Riemann-Roch theorem . 57 4.2 Application and Examples . 59 Chapter 5 Complex tori 65 5.1 Cohomology of complex tori . 66 5.2 Line bundles on complex tori . 67 1 5.3 Theta functions . 72 Chapter 6 The Jacobian Variety 77 6.1 Motivation: Abelian integrals . 77 6.2 Properties of the Jacobian variety . 79 6.3 Riemann’s theorem . 83 Chapter 7 The Torelli theorem 91 7.1 Proof of the Torelli theorem . 91 Chapter 8 Concluding remarks 98 8.1 The Schottky problem . 98 Chapter 9 Background material 100 9.1 Group cohomology . 100 9.2 Major theorems . 101 2 Acknowledgements I would like to extend my gratitude to all those who helped in the production of this thesis, and to those who supported me throughout this year.
    [Show full text]
  • Lectures on Torelli Theorems Spring School Rennes 2014
    Lectures on Torelli Theorems Spring School Rennes 2014 Chris Peters Universit´ede Grenoble I and TUE Eindhoven April 2014 Summary Lecture 1: Andreotti's classical beautiful proof of the Torelli theorem for curves (dating from 1958). The lecture ends with variational techniques due to Carlson, Green, Griffiths and Harris (1980), the curve case serving as a model case. Lecture 2 and 3 : A proof of the Torelli theorem for projective K3{surfaces modeled on the original proof of Piateˇckii-Shapiroand Safareviˇc(1971)ˇ but using the approach in the K¨ahlercase as given by Burns and Rapoport (1975), with modifications and simplifications by Looijenga and Peters (1981). If time allows for it I shall briefly point out some related developments. I particularly want to say something about derived Torelli and also about Verbitsky's recent proof of Torelli for hyperk¨ahlermanifolds. Contents 1 Torelli for Curves 2 1.1 What You Should Know . .2 1.2 Statements . .5 1.3 Andreotti's Proof . .5 1.4 A Variational Version . .7 2 Torelli for K3 Surfaces 9 2.1 Topology . .9 2.2 The Ample Cone . 10 2.3 Infinitesimal Torelli . 11 2.4 Global Torelli: Statement . 12 2.5 Global Torelli: Sketch of Proof . 14 1 3 Other Torelli Theorems 15 3.1 Infinitesimal Torelli . 15 3.2 Variational Torelli . 16 3.3 Torelli for Hyperk¨ahlerManifolds . 16 Introduction These Lectures are intended for (prospective) PhD-students with some back- ground in complex algebraic geometry. The first Lecture should be accessible to many since very little is as- sumed. It mainly explains Andreotti's classical beautiful proof [An] of the Torelli theorem for curves (dating from 1958).
    [Show full text]
  • Torelli Theorem for Stable Curves
    J. Eur. Math. Soc. *, 1–41 c European Mathematical Society 201* ! Lucia Caporaso Filippo Viviani · Torelli theorem for stable curves Received October 8, 2009 and in revised form July 13, 2010 Abstract. We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers. Keywords. Torelli map, Jacobian variety, theta divisor, stable curve, stable semi-abelic pair, com- pactified Picard scheme, semiabelian variety, moduli space, dual graph Contents 1. Introduction . 2 1.1. Problems and results . 2 1.2. Preliminaries . 4 1.3. First reductions . 9 2. Statement of the main theorem . 11 2.1. C1-equivalence . 11 2.2. Some properties of C1-sets . 13 2.3. Gluing points and gluing data . 14 2.4. Dual graphs of C1-equivalent curves . 16 3. T-equivalence: a second version of the Torelli theorem . 18 3.1. Definition of T-equivalence . 18 3.2. C1-equivalence equals T-equivalence . 21 4. Proof of the Main Theorem . 23 4.1. Combinatorial preliminaries . 23 4.2. Recovering gluing points from the theta divisor . 26 4.3. Torelli theorem: proof of the necessary condition . 29 4.4. Torelli theorem: proof of the sufficient condition . 32 5. The fibers of the Torelli morphism . 34 5.1. Injectivity locus and fiber cardinality of the Torelli morphism . 34 5.2. Dimension of the fibers . 39 References .
    [Show full text]
  • TORELLI THEOREM for the MODULI SPACES of RANK 2 QUADRATIC PAIRS 3 Determinant in the Usual Way
    TORELLI THEOREM FOR THE MODULI SPACES OF RANK 2 QUADRATIC PAIRS ANDRE´ G. OLIVEIRA Abstract. Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2. 1. Introduction Let X be a smooth projective curve over C. This paper deals with moduli spaces of quadratic pairs (V,γ) consisting of a vector bundle V together with a non-zero bilinear symmetric map γ on V , with values in a fixed line bundle U. The semistability condition for these objects depends on a real parameter α. Let Λ Nα denote the moduli space of α-polystable U-quadratic pairs (V,γ) with rk(V ) = 2 and deg(V ) = d, where the determinant of V is fixed to be Λ ∈ Jacd(X). The goal of this paper is to give new insights Λ on the geometry and topology of Nα . The main result is a proof that, under some conditions, a Torelli Λ type theorem holds for Nα — see Theorem 5.9 for the precise statement. We follow the methods used by Mu˜noz in [10], with the help of the results of [9]. One of the differences is that, in our case, the moduli spaces are not smooth. The fact that γ is a non-linear object (as opposed to most studied cases of vector bundles with extra structure) is perhaps one of the reasons why these moduli spaces have not yet been much studied. In fact, this non-linearity of γ is in the origin of many difficulties which arise, starting with the general definition of stability.
    [Show full text]