Vanishing Theorems for constructible Sheaves on Abelian Varieties

TL;DR

This paper proves vanishing theorems for hypercohomology of twisted perverse sheaves on complex abelian varieties, utilizing Tannakian formalism to analyze convolution products and Tannaka groups.

Contribution

It introduces new vanishing theorems for constructible sheaves on abelian varieties and develops a Tannakian framework for understanding their convolution structures.

Findings

Most character twists of perverse sheaves have vanishing hypercohomology in non-zero degrees.

Established a vanishing theorem for constructible sheaves on abelian varieties.

Analyzed properties of Tannaka groups arising from convolution of perverse sheaves.

Abstract

We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.