Liquid Tannaka Duality I: Classical Case

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.