A differential graded model for derived analytic geometry

TL;DR

This paper develops a new framework for derived analytic geometry using differential graded algebras with entire functional calculus, unifying complex and non-Archimedean settings for shifted Poisson structures and quantizations.

Contribution

It introduces a differential graded model for derived analytic geometry that aligns with existing theories in complex and non-Archimedean contexts, enabling advanced geometric and algebraic structures.

Findings

Recovers derived analytic spaces and stacks from Lurie's structured topoi in complex settings

Establishes a comparison between complex and non-Archimedean derived analytic geometries

Provides a foundation for shifted Poisson structures and quantizations in derived analytic geometry

Abstract

We give a formulation for derived analytic geometry built from commutative differential graded algebras equipped with entire functional calculus on their degree 0 part, a theory well-suited to developing shifted Poisson structures and quantisations. In the complex setting, we show that this formulation recovers equivalent derived analytic spaces and stacks to those coming from Lurie's structured topoi. In non-Archimedean settings, there is a similar comparison, but for derived dagger analytic spaces and stacks, based on overconvergent functions.