Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians PDF
Preview Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians
Encyclopaedia of Mathematical Sciences Volume 36 Editor-in-Chief: R. V. Gamkrelidze Springer-Verlag Berlin Heidelberg GmbH A. N. Parshin I. R. Shafarevich (Eds.) Algebraic Geometry III Complex Algebraic Varieties Algebraic Curves and Their J acobians Springer Consulting Editors of the Series: A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko, N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko Title of the Russian original edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental'nye napravleniya, VoI. 36, Algebraicheskaya Geometriya-3 Publisher VINITI, Moscow 1989 Mathematics Subject Classification (1991): 14-02, l4C20, 14C2l, l4C30, l4C34, 14D05, 14D07, 14D20, 14D22, 14F35, 32G20, 14H42, 14H45 ISSN 0938-0396 ISBN 978-3-642-08118-7 ISBN 978-3-662-03662-4 (eBook) DOI 10.1007/978-3-662-03662-4 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of iIIustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 Softcover reprint of the hardcover 1s t edition 1998 Typesetting: Camera-ready copy produced from the translator's input files using a Springer TEX macro package. Typesetting of diagrams by the translator using Paul Taylor's macros Commutative Diagrams in TeX. SPIN 10015170 4113143 -543210 - Printed on acid-free paper List of Editors, Authors and Translators Editor-in-Chief R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow; Institute for Scientific Information (VINITI), ul. Usievicha 20a, 125219 Moscow, Russia; e-mail: [email protected] Consulting Editors A. N. Parshin, I. R. Shafarevich, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia Authors Viktor S. Kulikov, Chair of Applied Mathematics II, Moscow State University of Transport Communications (MilT), ul. Obraztcova 15, 101475 Moscow, Russia; e-mail: [email protected] and [email protected] P.F. Kurchanov, Chair of Applied Mathematics II, Moscow State University of Transport Communications (MilT), ul. Obraztcova 15, 101475 Moscow, Russia V.V. Shokurov, Department of Mathematics, The Johns Hopkins University, Baltimore, MA 21218-2689, USA; e-mail: [email protected] Translator I. Rivin, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom, e-mail: [email protected]; Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA, e-mail: [email protected] Contents I. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures Vik. S. Kulikov, P. F. Kurchanov 1 TI. Algebraic Curves and Their Jacobians V. V. Shokurov 219 Index 263 I. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures Vik. S. Kulikov, P. F. Kurchanov Translated from the Russian by Igor Rivin Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 1. Classical Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 §1. Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 §2. Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 §3. A Comparison Between Algebraic Varieties and Analytic Spaces . 19 §4. Complex Manifolds as c= Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 24 §5. Connections on Holomorphic Vector Bundles . . . . . . . . . . . . . . . . . . 28 §6. Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 §7. Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 §8. Line Bundles and Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 §9. The Kodaira Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 §10. Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 2. Periods of Integrals on Algebraic Varieties . . . . . . . . . . . . . . . 66 §1. Classifying Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 §2. Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 §3. The Period Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 §4. Variation of Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 V. S. Kulikov et al., Algebraic Geometry III © Springer-Verlag Berlin Heidelberg 1998 2 Vik. S. Kulikov, P. F. Kurchanov §5. Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 §6. Infinitesimal Variation of Hodge Structures . . . . . . . . . . . . . . . . . . . . 97 Chapter 3. Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 §1. Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 §2. The Cubic Threefold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 §3. K3 Surfaces and Elliptic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 §4. Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 §5. Counterexamples to Torelli Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 140 Chapter 4. Mixed Hodge Structures .............................. 143 §1. Definition of mixed Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . 143 §2. Mixed Hodge structure on the Cohomology of a Complete Variety with Normal Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 §3. Cohomology of Smooth Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 §4. The Invariant Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 §5. Hodge Structure on the Cohomology of Smooth Hypersurfaces ... 168 §6. Further Development of the Theory of Mixed Hodge Structures . . 176 Chapter 5. Degenerations of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . 183 §1. Degenerations of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 §2. The Limit Hodge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 §3. The Clemens-Schmid Exact Sequence ........................ 190 §4. An Application of the Clemens-Schmid Exact Sequence to the Degeneration of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 §5. An Application of the Clemens-Schmid Exact Sequence to Surface Degenerations. The Relationship Between the Numerical Invariants of the Fibers Xt and X0. . . . . . . . . . . . . . . . . . . . . . . . . . . 199 §6. The Epimorphicity of the Period Mapping for K3 Surfaces 205 Comments on the bibliography 211 References 213 Index ......................................................... 261 Periods of Integrals and Hodge Structures 3 Introduction Starting with the end of the seventeenth century, one of the most interesting directions in mathematics (attracting the attention as J. Bernoulli, Euler, Jacobi, Legendre, Abel, among others) has been the study of integrals of the form l = r -dz, Aw(T) TO W where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation (1) where the polynomial on the right hand side has no multiple roots. In this case the function Aw is called an elliptic integral. The value of Aw is determined up to mv1 + nv2, where v1 and v2 are complex numbers, and m and n are integers. The set of linear combinations mv1 + nv2 forms a lattice H C C, and so to each elliptic integral Aw we can associate the torus C/ H. On the other hand, equation (1) defines a curve in the affine plane C2 = {(z,w)}. Let us complete C2 to the projective plane lP'2 = lP'2(C) by the addition of the "line at infinity", and let us also complete the curve defined by equation (1). The result will be a nonsingular closed curve E C lP'2 (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve. C/ It is a remarkable fact that the curve E and the torus H are isomorphic Riemann surfaces. The isomorphism can be given explicitly as follows. Let p(z) be the Weierstrass function associated to the lattice H C C. 1 L [ 1 1 ] p = z2 + (z- 2h)2 - (2h)2 • hEH,h=/-0 It is known that p(z) is a doubly periodic meromorphic function with the period lattice H. Further, the function p(z) and its derivative p'(z) are related as follows: (2) for certain constants g2 and g3 which depend on the lattice H. Therefore, the mapping z --+ (p( z), p' ( z)) is a meromorphic function of Cf H onto the compactification E' C lP'2 of the curve defined by equation (2) in the affine plane. It turns out that this mapping is an isomorphism, and furthermore, the projective curves E and E' are isomorphic! Let us explain this phenomenon in a more invariant fashion. The projection (z, w) --+ z of the affine curve defined by the equation (1) gives a double 4 Vik. S. Kulikov, P. F. Kurchanov covering 1r : E -t IP'1, branched over the three roots z1, z2, z3 of the polynomial z3 + pz + q and the point oo. The differential w = dz/2w, restricted to E is a holomorphic 1-form (and there is only one such form on an elliptic curve, up to multiplication by con coo stants). Viewed as a manifold, the elliptic curve E is homeomorphic to the product of two circles S1 x Sl, and hence the first homology group H1 (E, Z) is isomorphic to Z E9 Z. Let the generators of H1(E,Z) be ')'1 and ')'2. The w} . lattice H is the same as the lattice { m f-r w + n J'Y Indeed, the elliptic 1 2 integral Aw is determined up to numbers of the form fz .j 1 , where l is z3+pz+q a closed path in C\ {z 1 , z2, Z3}. On the other hand l i J z3 +1 pz + q = w, where ')' is the closed path in E covering l twice. J . The integrals w are called periods of the curve E. The lattice H is called the period lattice.1fhe discussion above indicates that the curve E is uniquely determined by its period lattice. This theory can be extended from elliptic curves (curves of genus 1) to curves of higher genus, and even to higher dimensional varieties. Let X be a compact Riemann surface of genus g (which is the same as a nonsingular complex projective curve of genus g). It is well known that all Riemann surfaces of genus g are topologically the same, being homeomorphic to the sphere with g handles. They may differ, however, when viewed as complex analytic manifolds. In his treatise on abelian functions (see de Rham [1955]), Riemann constructed surfaces (complex curves) of genus g by cutting and pasting in the complex plane. When doing this he was concerned about the periods of abelian integrals over various closed paths. Riemann called those periods (there are 3g- 3) moduli. These are continuous complex parameters which determine the complex structure on a curve of genus g. One of the main goals of the present survey is to introduce the reader to the ideas involved in obtaining these kinds of parametrizations for algebraic varieties. Let us explain this in greater detail. On a Riemann surface X of genus g there are exactly g holomorphic 1- forms linearly independent over C. Denote the space of holomorphic 1-forms on X by H1•0, and choose a basis w = (w1, ... ,wg) for H1•0. Also choose a basis/'= (/'1, ... ,')'2g) for the first homology group H1(X,Z) ~ Z2g. Then the numbers are called the periods of X. They form the period matrix Jl = ( Jlii). This matrix obviously depends on the choice of bases for H1•0 and H1(X,Z). It turns out (see Chapter 3, Section 1), that the periods uniquely determine the curve X. More precisely, let X and X' be two curves of genus g. Suppose
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