Families of abelian varieties over curves with maximal Higgs field

Abstract

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the Arakelov inequalities say that 2deg(F(1,0)) is bounded from above by g=rank(F(1,0))(2q-2+s). We show that for s>0 families reaching this bound are isogenous to the g-fold product of a modular family of elliptic curves, and a constant abelian variety. The content of this note became part of the article "A characterization of certain Shimura curves in the moduly stack of abelian varieties" (math.AG/0207228), where we also handle the case s=0.