Lectures on Analytic Geometry

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.