On subvarieties of abelian varieties

Abstract

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, BcX 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-QQ~Q-QQ I Q/R 1 IQQ~/QA~Q/~Q~ ?Q

On a Theorem of Ziv Ran concerning Abelian Varieties Which Are Product of Jacobians

“…The fact that = = 1 for all , is Corollary 10. For the other points, the proof follows closely the one from [3] with some modifications of the arguments. We began with three preliminary steps.…”

“…Some points are as in [3] and are included only for the sake of completeness. The modifications appear from the replacement of Lemma II.8 from [3] with the result below whose proof is very simple.…”

“…[2]), Matsusaka and Hoyt gave a necessary and sufficient criterion for an Abelian variety for being a Jacobian, respectively, a product of Jacobians. In [3], Ran reconsiders the subject and gives a more general and probably more natural criterion for this. However, his method seems unsatisfactory in positive characteristic.…”

A new proof of the Ran-Matsusaka criterion for Jacobians

“…Ran's criterion [3] is an extension of Matsusaka's criterion [1]. Ran points out that Matsusaka's main tool is a certain endomorphism a; the composite map f b*r in the diagram below is the endomorphism a(G, D) of [1].…”

“…Following the referee's advice we omit it because it is a rather straightforward exercise, cf. p. 469 in [3].…”