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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.
|Supervisor:||Weissauer, Prof. Dr. Rainer|
|Date of thesis defense:||13 June 2013|
|Date Deposited:||26 Jun 2013 08:55|
|Faculties / Institutes:||The Faculty of Mathematics and Computer Science > Department of Mathematics|
|Controlled Keywords:||Algebraische Geometrie|
|Uncontrolled Keywords:||Tannakian Categories, Perverse Sheaves, Abelian Varieties, Convolution, Vanishing theorems, Theta divisor|