Classification of nonrigid families of abelian varieties

Abstract

We will give a complete classification of non-rigid families of abelian varieties by means of the endomorphism algebra of the variation of Hodge structure. As a consequence, we can obtain several conditions of rigidity for abelian schemes. For example, we show that an abelian scheme which has no isotrivial factor is rigid if the relative dimension is less than 8. Moreover, examples of non-rigid abelian schemes are obtained as Kuga fiber spaces associated to symplectic representations classified by Satake. Introduction. Let Y be an algebraic curve defined over an algebraically closed field k of characteristic zero, and let Σ a Y be a finite set of points. Faltings [F] has shown a theorem of Arakelov-type for abelian varieties, that is, there are only finitely many families of principally polarized abelian varieties of relative dimension g on Y, with good reduction outside Σ, and satisfying the condition (*) in [F], His proof consists of two ingredients. First he showed that the moduli space of families of principally polarized abelian varieties on Y with good reduction outside Σ is a scheme of finite type over k (a boundedness result). Next he proved that a family of abelian varieties cannot be deformed (i.e., a family is rigid) if and only if the condition (*) is satisfied. The condition (*) says essentially that all endomorphisms of the local system of the first (co-)homology groups of fibers come from endomorphism of the abelian varieties, and Deligne [D] has shown that the condition is satisfied by a family of abelian varieties which has no isotrivial factors and the relative dimension <3. On the other hand, following Deligne's suggestion, Faltings [F] gave an example of non-rigid families of abelian varieties with relative dimension 8 which has no isotrivial factors. So it is interesting to ask, for example, whether there exists a non-rigid family of abelian varieties of relative dimension d, 4<d<Ί9 which has no isotrivial factors. In this paper, we will give a complete classification of non-rigid families of abelian varieties by means of the endomorphism algebra of the variation of Hodge structure of the first homology (or cohomology) groups of the fibers. Let S be a connected smooth quasi-projective variety over C, and / : X-+S an * Supported in part by the Japan Foundation and JAMI of the Johns Hopkins University. 1991 Mathematics Subject Classification. Primary 14J10; Secondary 14G35, 14G40.