Liquid Tannaka Duality I: Classical Case

TL;DR

This paper establishes a Tannaka duality for analytic stacks using liquid vector spaces, enabling the reconstruction of fundamental groups and flat connections from categorical data.

Contribution

It introduces a Tannaka duality framework for geometric stacks modeled on Stein spaces utilizing liquid vector spaces, a novel approach in the field.

Findings

Reconstructed the topological fundamental group from liquid local systems.

Reconstructed twisted fundamental groupoids related to meromorphic flat connections.

Applied liquid Tannaka duality to complex algebraic varieties and analytic stacks.

Abstract

We prove a Tannaka duality statement for geometric stacks in the setting of analytic stacks modelled on globally finitely presented Stein spaces. The key ingredient is the theory of liquid vector spaces and liquid quasicoherent sheaves of Clausen-Scholze. As an application, we reconstruct the topological fundamental group of any complex algebraic variety from its category of liquid local systems. We also reconstruct a series of "twisted fundamental groupoids" whose representations correspond to meromorphic flat connections on the complex affine line with logarithmic or irregular singularities at the origin.