Liquid Tannaka Duality I: Classical Case
Waleed Qaisar, Gregory Taroyan
TL;DR
The paper establishes a Tannaka duality for geometric analytic stacks modeled on globally finitely presented Stein spaces by leveraging liquid vector spaces and liquid quasicoherent sheaves. It extends Lurie-style duality into the analytic realm via Clausen–Scholze liquid theory and animated EFC algebras, enabling holomorphic pushforwards and a rigorous infinitesimal criterion for smoothness. As applications, it reconstructs the topological fundamental group of complex varieties and a hierarchy of Stokes groupoids from liquid local systems, and it recovers Stein complex Lie groups from their liquid representations. It also demonstrates the essential role of quasicoherent sheaves in analytic Tannaka duality and outlines future directions toward broader analytic stacks and C∞-settings.
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.