On subvarieties of abelian varieties

Abstract

Let A be a complete nonsingular algebraic curve of genus n over IF. The Jacobian J(A) has the standard subvarieties Wj={a I +...ae: areA}, considering A as embedded in J(A), and PoincarCs formula tells us that the cohomology class [Wj=[O]"-e/(n-d)!, where O is the canonical principal polarization of J(A). Thus [We] is a primitive (indivisible) positive class. The question we will be concerned with in this paper is, roughly speaking, to what extent are such cohomology classes characteristic of W~ subvarieties in Jacobians among subvarieties of abelian varieties. The first result in this direction is due to Matsusaka [6] who gave the following criterion: Let X be an abelian variety of dimension n, B c X a divisor giving a principal polarization, A c X an effective l-cycle homologous to B"1/(n 1 ) ! ; then X is the Jacobian J(A), and, up to a translation, A is canonically embedded in X, and B is the canonical theta-divisor on J(A) (we will abbreviate this conclusion by saying that (X, B, A) is a Jacobian triple). Here we shall prove the following refinement of Matsusaka's criterion: if B c X is any ample divisor, A c X an effective 1-cycle generating X such that the intersection number A. B = n, the smallest possible value, then (X, B, A) is a Jacobian triple. A partial extension of Matsusaka's theorem was obtained by Barton and Clemens [1] who proved, in dimension four, that the locus of principally polarized abelian varieties (X, O) carrying a subvariety homologous to 02/2 is a proper subset of the moduli space. Here we generalize this result by proving that the locus of (X,O) of dimension n carrying subvarieties A, B homologous, respectively, to Od/d! and O" e/(n-d)! contains the Jacobians as an irreducible component. In fact, we prove more precisely that any deformation of a triple (J(C), W,_ e, We), where C is nonhyperelliptic, must be induced by a deformation of C. Our strongest results are in dimension four, where we prove that W 2 subvarieties in Jacobians are characterized among surfaces in abelian fourfolds by having the smallest possible self-intersection number, without being "degenerate". We then apply this result to the Schottky problem, explicitly characterizing Jacobians among principally polarized abelian fourfolds (X,O), essen-