Lectures on Analytic Geometry

TL;DR

This paper presents updated lecture notes on analytic geometry, covering liquid real vector spaces and a tentative category of analytic spaces, with relevance to complex and adic spaces.

Contribution

It introduces a new approach to analytic geometry through liquid real vector spaces and discusses a tentative category of analytic spaces.

Findings

Developed the theory of liquid real vector spaces.

Proposed a tentative definition of a category of analytic spaces.

Discussed the relevance of these concepts to complex and adic spaces.

Abstract

This is a slightly updated version of lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. This is intended as a stable citable version of the material. In the first half of this course, we develop the basic theory of liquid real vector spaces, which we used in another course to give a new approach to complex-analytic geometry. In the second half, we gave a tentative definition of a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces. While the precise definition of analytic spaces represents an abandoned stepping stone on our way to define analytic stacks and hence should be seen as a historical artifact, much of the surrounding discussion stays very relevant.