Condensed Mathematics and Complex Geometry

TL;DR

This paper revisits classical complex-analytic geometry using condensed mathematics, providing alternative proofs for key theorems like Serre duality and GAGA without introducing new geometric concepts.

Contribution

It offers a new perspective on complex geometry by redeveloping classical theorems through condensed mathematics, emphasizing a different foundational approach.

Findings

Reproves finiteness of coherent cohomology for compact complex manifolds.

Provides alternative proofs of Serre duality and GAGA.

Reestablishes the Hirzebruch--Riemann--Roch theorem in this framework.

Abstract

This is a slightly revised version of lectures notes for a course in Summer 2022 joint between Bonn and Copenhagen, intended as a stable citable version. The goal of this course is to make our general approach to analytic geometry via condensed mathematics more concrete by concentrating on the case of complex-analytic geometry. Instead of trying to develop new kinds of geometry, here we only try to redevelop the classical theory from a different point of view. More precisely, we reprove some important theorems for compact complex manifolds, including finiteness of coherent cohomology, Serre duality, GAGA and (Grothendieck--)Hirzebruch--Riemann--Roch.