On the Tannaka group attached to the Theta divisor of a generic principally polarized abelian variety

摘要

To any closed subvariety Y of a complex abelian variety one can attach a reductive algebraic group G which is determined by the decomposition of the convolution powers of Y via a certain Tannakian formalism. For a theta divisor Y on a principally polarized abelian variety, this group G provides a new invariant that naturally endows the moduli space of principally polarized abelian varieties of dimension g with a finite constructible stratification. We determine G for a generic principally polarized abelian variety, and for we show that the stratification detects the locus of Jacobian varieties inside the moduli space of abelian varieties.