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arXiv · 1309.3754

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

Thomas Krämer·Rainer Weissauer
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Abstract

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 $A_g$ 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 $g=4$ we show that the stratification detects the locus of Jacobian varieties inside the moduli space of abelian varieties.

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Thomas Krämer, Rainer Weissauer. 2013-09-15. On the Tannaka group attached to the Theta divisor of a generic principally polarized abelian variety. https://doi.org/10.1007/s00209-015-1505-9

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