Tannaka Duality for Geometric Stacks

TL;DR

This paper establishes a Tannaka duality framework for algebraic stacks, linking morphisms to tensor functors, and demonstrates that certain analytic maps are algebraic under specific conditions.

Contribution

It extends Tannaka duality to geometric stacks and proves that analytic maps from proper complex varieties are algebraic.

Findings

Identification of morphism groupoids with tensor functor categories

Proof that analytic maps from proper varieties are algebraic

Extension of Tannaka duality principles to algebraic stacks

Abstract

We show that, under appropriate hypothesis, the groupoid of maps from S to an an algebraic stack X can be identified with a category of tensor functors from coherent sheaves on X to coherent sheaves on S. As an application, we show that if S is a proper variety over the field of complex numbers, then every ``analytic'' map from S to X is ``algebraic''.