Berkovich analytification and tropicalization

Dr. Daniele Turchetti (University of Warwick)

Tuesdays, 13:15 to 15:00, HG G 43
20 September to 6 December 2022

Abstract

The course is an introduction to the analytification and tropicalization techniques that recently led to breakthroughs in many topics of algebraic geometry. These constructions associate with an algebro-​geometric object a "combinatorial shadow" (tropicalization) and an "analytic enhancement" (Berkovich analytification), which are tightly related to each other.

These techniques were put into use to solve longstanding open problems in a wide range of subjects: Harris-​Taylor's proof of the local Langlands conjecture for GL_n, Chan-​Galatius-Payne's computation of the top weight cohomology of moduli spaces of curves, and Nicaise-​Xu-Yu's construction of a non-​Archimedean SYZ fibration in the framework of mirror symmetry, just to name a few.

The course is divided in three parts, each consisting of 3-4 lectures. There will be many examples and comparisons between the two constructions, to show how they relate to each other. No previous background in non-​archimedean or tropical geometry is assumed, but basic knowledge of algebraic geometry is helpful to understand the topic.

1. Berkovich geometry and analytification: in this part, I introduce Berkovich analytic spaces over non-​Archimedean fields; I give an overview of their nice topological and analytic properties; and I illustrate how to associate with an algebraic variety over a non-​Archimedean field its analytification.
2. Tropical geometry and tropicalization: in this part, I define tropical varieties and describe their polyhedral structure; I explain how to associate a tropical variety to an algebraic variety and show a few straightforward applications of tropical methods in the case of curves; the final lectures are devoted to highlight the relationship between tropicalization and analytification, via the notion of "skeleton" of a Berkovich space and a theorem by S. Payne giving a topological description of analytifications as "limits of tropicalizations".
   
3. Applications: these might vary depending on the interests of the participants and will be decided together in the first days. Options include: tropical Brill-​Noether theory, the top weight cohomology of M_{g,n}, tropical techniques for counting covers of curves with prescribed ramification, essential skeletons with applications to birational geometry, and hybrid Berkovich spaces.