Families of abelian varieties over curves with maximal Higgs field

Details Families of abelian varieties over curves with maximal Higgs field Families of abelian varieties over curves with maximal Higgs field

2002-04-22

arxiv.org/abs/math/0204261v2

Mathematics

Algebraic Geometry

13 pages, Latex, two minor errors corrected, the content of this note became part of math.AG/0207228

Scientific paper

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.

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