Welcome!

This course will be an introduction to definable complex analytic spaces, which are spaces locally modeled on zero sets of complex analytic functions which are definable in a fixed o-minimal structure. There will be roughly three parts:

• First we will develop the basic theory of definable complex analytic spaces, which will run parallel to that of algebraic schemes and complex analytic spaces except that a very mild Grothendieck topology must be used in the sheaf theory. Thus, this part of the course will serve both as a refresher for these geometries as well as a template for more nuanced ones, such as algebraic spaces and rigid analytic spaces.

• We will then discuss algebraization theorems, culminating in the definable GAGA theorem.

• Finally, we will detail applications to flat vector bundles, Hodge theory, and transcendence theory.