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Algebraic Geometry III Complex Algebraic Varieties Algebraic Curves and Their Jacobians A. N. parshin I. R. Shafarevich (Eds.) Algebraic Geometry III Complex Algebraic Varieties Algebraic Curves and Their Jacobians Springer 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 (MIIT), 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 (MIIT), 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 9 1125, USA, e-mail: [email protected] Contents I. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures Vik. S. Kulikov, P. F. Kurchanov 1 II. 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 53. A Comparison Between Algebraic Varieties and Analytic Spaces _ 19 $4. Complex Manifolds as C” Manifolds ......................... 24 $5. Connections on Holomorphic Vector Bundles .................. 28 56. Hermitian Manifolds ........................................ 33 $7. Kahler Manifolds .......................................... 38 58. Line Bundles and Divisors ................................... 49 59. The Kodaira Vanishing Theorem ............................. 54 §lO.Monodromy ............................................... 60 Chapter 2. Periods of Integrals on Algebraic Varieties ............... 66 $1. Classifying Space .......................................... 66 $2. ComplexTori .............................................. 77 $3. The Period Mapping ....................................... 84 $4. Variation of Hodge Structures ............................... 88 2 Vik. S. Kulikov, P. F. Kurchanov 55. 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 85. Counterexamples to Torelli Theorems . 140 Chapter 4. Mixed Hodge Structures . 143 31. 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 52. 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 Xe. .......................... 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 A,(7) = JT; $, where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation w2 =2+pz+q, where the polynomial on the right hand side has no multiple roots. In this case the function A, is called an elliptic integral. The value of A, is determined up to mvl + nus, where ~1 and u2 are complex numbers, and m and n are integers. The set of linear combinations my +nv2 forms a lattice H c Cc, and so to each elliptic integral A, we can associate the torus C/H. On the other hand, equation (1) defines a curve in the affine plane C2 = {(z, w)}. Let us complete C? to the projective plane P2 = lF’2(C) by the addition of the “line at infinity”, and let us also complete the curve defined by equation (1). The result will be a nonsingular closed curve E c lP2 (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve. It is a remarkable fact that the curve E and the torus C/H are isomorphic Riemann surfaces. The isomorphism can be given explicitly as follows. Let p(z) be the Weierstrass function associated to the lattice H c C!. -- *=$+ hEH,h#Oc 1(z -l2h)2 (2$ I . 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: (d)” = 4v3 - g2k3 - 93, (2) for certain constants gs and gs which depend on the lattice H. Therefore, the mapping 2 + (dz),d(~)) is a meromorphic function of C/H onto the compactification E’ c IP2 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 7r : E + p’, branched over the three roots ~1, ~2,~s of the polynomial z3 + pa + q and the point 00. The differential w = dz/2w, restricted to E is a holomorphic l-form (and there is only one such form on an elliptic curve, up to multiplication by con- stants). Viewed as a C” manifold, the elliptic curve E is homeomorphic to the product of two circles S1 x S1, and hence the first homology group Hr (E, Z) is isomorphic to Z @ Z. Let the generators of Hr (E, Z) be yr and 72. The lattice H is the same as the lattice {m ST1w + n. ST2w} . Indeed, the elliptic integral A, is determined up to numbers of the form sl J&, where 1 is a closed path in e\{ zl, ~2, zs}. On the other hand where y is the closed path in E covering 1 twice. The integrals s w are called periods of the curve E. The lattice H is called the period lattice. ?‘h e d’rscussion 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 9). 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) mod&. 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 l- forms linearly independent over @. Denote the space of holomorphic l-forms on X by H1lO, and choose a basis w = (WI,. , wg) for H1>‘. Also choose a basis y = (71,. , ~2~) for the first homology group HI (X, Z) E E2g. Then the numbers f2ij = Wi s Yj are called the periods of X. They form the period matrix R = (0,j). This matrix obviously depends on the choice of bases for H1>’ and Hl(X, Z). It turns out (see Chapter 3, Section l), that the periods uniquely determine the curve X. More precisely, let X and X’ be two curves of genus g. Suppose Periods of Integrals and Hodge Structures 5 w and w’ are bases for the spaces of holomorphic differentials on X and X’, respectively, and y and y’ be are bases for Hr (X, Z) and Hi (X’, Z) such that there are equalities (%.“lj)X = (rh;)X~ between the intersection numbers of y and 7’. Then, if the period matrices of X and X’ with respect to the chosen bases are the same, then the curves themselves are isomorphic. This is the classical theorem of Torelli. Now, let X be a non-singular complex manifold of dimension d > 1. The complex structure on X allows us to decompose any complex-valued C” differential n-form w into a sum p+q=n of components of type (p, Q). A form of type (p, q) can be written as wP,q = A dZip A d.Zjl A . A dzj,. c hl,Jdai, A . (I,J)=(il,...,ip,jl,...,jq) If X is a projective variety (and hence a Kahler manifold; see Chapter 1, Section 7) , then this decomposition transfers to cohomology: H”(X,a-J = @ IPq, HfJ~Q= iPIP.
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