Families of abelian varieties over curves with maximal Higgs field
Abstract
Let f:X>Y be a semistable 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 nonflat (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))(2q2+s). We show that for s>0 families reaching this bound are isogenous to the gfold 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2002
 DOI:
 10.48550/arXiv.math/0204261
 arXiv:
 arXiv:math/0204261
 Bibcode:
 2002math......4261V
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 14K10 (Primary) 14D05;
 14D07 (Secondary)
 EPrint:
 13 pages, Latex, two minor errors corrected, the content of this note became part of math.AG/0207228