Higher analytic stacks and GAGA theorems

TL;DR

This paper develops the theory of higher geometric stacks in complex and non-archimedean analytic geometry, establishing foundational results and GAGA theorems using infinity categories to simplify the framework.

Contribution

It introduces a new foundation for higher stacks in analytic geometry and proves GAGA theorems in this context, extending classical results to higher categorical settings.

Findings

Proved analog of Grauert's theorem for derived direct images

Established GAGA theorems for higher stacks

Simplified theory using infinity categories

Abstract

We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-\'etale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case.