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Lectures on Torelli Theorems Spring School Rennes 2014

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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). It ends with a section mak- ing the bridge to a more recent approach, namely the one using variational techniques. The curve case serves as a model case for these techniques. This approach is due to Carlson, Green, Griffiths and Harris [CaGGH] (1980). The second Lecture is much more demanding but this is compensated by giving ample references. I explain here a proof of the Torelli theorem for pro- jective K3{surfaces modeled on the original proof of Piateˇckii-Shapiroand Safareviˇc[Pi-S]ˇ (1971) but using the approach in the K¨ahlercase as given by Burns and Rapoport [B-R] (1975), with modifications and simplifications by Looijenga and Peters [L-P] (1981). The last Lecture is meant to complement the first two by briefly point- ing out some recent (and less recent) developments. I particularly want to mention the recent proof of Torelli for hyperk¨ahlermanifolds [V] due to Verbitsky. Acknowledgments I thank Hans Sterk whose remarks led to improvements in the presentation. 1 Torelli for Curves 1.1 What You Should Know Consult [G-H, Ch 2.7] and [A-C-G-H, Ch. 1]. The original result is [T]. By a curve we mean a smooth complex projective curve. Let C be a curve of genus g ≥ 2 and let 0 ∗ φK : C ! P(H (KC ) ) 0 be the canonical map. Concretely, if f!1;:::;!gg is a basis of H (KC ), g−1 φK (x) = (!1(x): ··· : !g(x)) 2 P : It is known that φK is biholomorphic onto its image for non-hyperelliptic C g−1 while, if C is hyperelliptic, φK is 2 to 1 onto a rational normal curve in P of degree g − 1. In that case φK is ramified in its 2g + 2 Weierstraß points. 2 Let x1; : : : xd 2 C and D = x1 + ··· + xd the corresponding divisor. Put hφK Di = proj. subspace spanned by φK (x1); : : : ; φK (xd): Since 1 0 h (D) = h (KC − D) = codimhφK Di; Riemann-Roch implies 0 dimhφK Di = deg(D) − h (D) (geometric form of Riemann-Roch): 0 In particular, for generic points xj one has h (D) = 1 and so the images φK (xj) span a projective subspace of maximal dimension d − 1. 1;0 0 Integrating along a 1-cycle γ gives a function on H (C) = H (KC ) which only depends on the homology class [γ] 2 H1(X; Z). This gives an in- 0 ∗ jective homomorphism ι : H1(C; Z) ,! H (KC ) ; by definition the quotient 0 ∗ J(C) = H (KC ) /ι(H1(C; Z)) is the Jacobian J(C). It is a complex torus of dimension g since Im(ι) 0 can be shown to be a lattice in H (KC ). This torus admits a projective embedding as we are now going to explain. 1 1 To start with, there is a canonical identification H (J(C); Z) = H (C; Z) 2 2 1 2 1 and so H (J(C); Z) = Λ H (J(C); Z) = Λ H (C; Z) which implies that 1 the cup product form Q on H (C; Z) can be viewed as an element q 2 2 H (J(C); Z). The cup product form obeys the (Riemann bilinear rela- tions): Z Q(!; !0) = ! ^ !0 = 0; i · Q(!; !¯) > 0 for ! 6= 0: (1) X These can be reinterpreted in terms of the Hodge decomposition 1 1;0 0;1 H (C; C) = H (C) ⊕ H (C) by saying that H1;0(C) is Q-isotropic and that the Hodge metric h(x; y) := 2 i!(x; y¯) is a positive definite metric. The Hodge decomposition on H (J(C); C) reads 2 2;0 1;1 0;2 H (J(C); C) = H (J(C)) ⊕ H (J(C)) ⊕ H (J(C)) = Λ2H1;0(C) ⊕ H1;0(C) ⊗ H0;1(C) ⊕ Λ2H0;1(C) 1;1 2 and then q 2 H (J(C)) \ H (X; Z) and q > 0. Hence (by Lefschetz' theorem on (1; 1)-classes and Kodaira's ampleness criterion), q = c1(Θ), where ΘC ⊂ J(C) is an ample divisor, the theta divisor. Such a class is called a polarization. Since q is unimodular, this polarization is special: it is a principal polarization. 3 Fix a point o 2 C. Recall the Abel-Jacobi map u : C ! J(C) Z x x 7! u(x) = integration functional ! 7! !: o It induces the morphisms u(d) : Cd ! J(C) (x1; : : : ; xd) = u(x1) + ··· + u(xd): In what follows I need the Jacobi matrix of u(d) (with respect to a basis 0 d f!1;:::;!gg of H (KC )) at the point x = (x1; ··· ; xd) 2 C : 0 1 !1(x1) ··· !g(x1) (d) B .. .. C J (u )x := @ . A : (2) !1(xd) ··· !g(xd) The image of the Abel-Jacobi map is denoted classically as (d) Wd := Im(u ): To simplify notation, I do not make a distinction between the point x and d the degree d divisor D = x1 + ··· xd and then u(D) means u (x) From (2) one sees: 0 ∗ (d) Tu(D)(Wd) = fsubspace of (H (KC ) ) spanned by the rows of J (u )xg: By the geometric form of the Riemann-Roch theorem one deduces: (d) Lemma 1.1. Let Wd be the image of u . Then Wd is smooth at the point u(D) if h0(D) = 1, and if this is the case, one has: PTu(D)Wd = hφK Di: Finally, recall some classical theorems involving the Abel-Jacobi map: Theorem 1.2. (1) u(D) = u(D0) if and only if D are D0 linearly equiva- lent; 0 (2) Wd is smooth at u(D) if and only if h (D) = 1; (g−1) (3) Wg−1, the image of u , is a translate of the theta-divisor. 4 1.2 Statements 0 Theorem 1.3 (Torelli). Let C; C be two genus g curves such that (J(C); ΘC ) ' 0 0 (J(C ); ΘC0 ), then C and C are isomorphic. Let us explain how this can be rephrased purely in terms of Hodge the- ory. Giving the Jacobian J(C) is the same as giving H1;0(C) together with the integral lattice ιH1(C; Z). The differentiable torus underlying J(C) is 1 1 1 just H (C; R)=H (C; Z). The Hodge structure on H (C) defines the Weil- 1 1 1;0 operator W : H (C; C) ! H (C; C) given by W jH = multiplication by i while W jH0;1(C) is multiplication by −i. This operator is real; hence an 1 1 2 operator W : H (C; R) ! H (C; R). Since W = −1, this is a complex structure. Hence J(C) is a complex torus. The cup product form Q on 1 H (C; Z) satisfying the Riemann bilinear relations (1) is exactly what is 1 called a polarization for the Hodge structure on H (C; Z). Summariz- ing: Corollary 1.4. A genus g curve is determined up to isomorphism by the 1 polarized Hodge structure (H (C; Z);Q). Torelli can also be reformulated in terms of the period map. Let Mg be the coarse moduli space of genus g curves. It is a quasi-projective variety of dimension 3g − 3 (provided g ≥ 2). A point of Mg represents a genus g curve, or, rather, a class [C] of a curve up to isomorphism. The coarse moduli space Ag of principally polarized abelian varieties of dimension g is 1 a quasi-projective variety of dimension 2 g(g + 1). The map [C] 7! [J(C)] induces a morphism of algebraic varieties p : Mg ! Ag, the period map, and Torelli is the statement: Corollary 1.5. The period map p : Mg ! Ag is an injective morphism. Remark. It does not follow that p is an immersion since Mg and Ag have singularities. Nevertheless, this is true [O-S] and goes by the name local Torelli theorem.
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