%0 Generic
%A Krämer, Thomas
%D 2013
%F heidok:15066
%K Tannakian Categories, Perverse Sheaves, Abelian Varieties, Convolution, Vanishing theorems, Theta divisor
%R 10.11588/heidok.00015066
%T Tannakian Categories of Perverse Sheaves on Abelian Varieties
%U https://archiv.ub.uni-heidelberg.de/volltextserver/15066/
%X We study Tannakian categories attached to perverse sheaves on abelian varieties with respect to the convolution product. The construction of these categories is closely intertwined with a cohomological vanishing theorem which is an analog of Artin's affine vanishing theorem and contains the generic vanishing theorems of Green and Lazarsfeld as a special case. To illustrate the geometric relevance of the developed notions, we determine the Tannaka group of the theta divisor on a general principally polarized complex abelian variety of arbitrary dimension and explain its relationship with the Schottky problem in genus 4. Here the convolution square of the theta divisor describes a family of surfaces of general type, and a detailed study of this family leads to a variation of Hodge structures with monodromy group W(E6) which has a natural interpretation in terms of the Prym map. In the final chapter we take a closer look at convolutions of curves inside Jacobian varieties and provide a recursive formula for the generic rank of Brill-Noether sheaves.