Time: 9:00-12:00, April 14/15/17/21/22/24/28/29
Venue: E4-233
Speaker: Antoine Ducros, Sorbonne Université
Abstract:
In this series of lectures, I will first give a general presentation of Berkovich's approach of non-archimedean geometry, give the basic definitions and the main general properties of Berkovich spaces, and illustrate the with a lot of examples, especially in dimension 1. Then I will devote some time to the deep links between Berkovich and tropical (or polyhedral) geometry, with a special focus on skeletons — skeletons are subsets of Berkovich spaces that inherit a natural polyhedral structure. After that, I will introduce the theory of real-valued differential forms and current on Berkovich spaces which we have been developing since almost 15 years with Antoine Chambert-Loir. The formalism we get is close to the classical complex formalism: we can define integrals of (n,n)-forms (with n the dimension of the ambient space) boundary integrals of (n,n-1)-forms, we have a Stokes formula… We can define currents, prove a Poincaré-Lelong formula, define the curvature form or current of a metrized line bundles, and in some cases, we can multiply curvature currents (following Bedford-Taylor's strategy) and get interesting measures on our Berkovich space, which are related to the intersection theory on the residue field. I will give detailed definitions of all the aforementioned objects, try to provide a lot of significant examples, state our main results, and explain their proofs, or at least their key steps and core arguments.
