Seminar Algebra and Geometry: Antoine Ducros (Université Paris 6)

Real differential forms and currents in p-adic geometry

If one tries to mimic naively in the  p-adic setting what is usually done in complex analytic geometry, one immediately faces big problems, because p-adic fields are totally disconnected. Hence in order to develop a relevant p-adic geometry, a more subtle approach is needed.

In this talk, I will begin with a presentation of that of Berkovich. It roughly consists of 'adding plenty of points' to the usual p-adic spaces so that they get good topological properties (like local path-connectedness). I will describe some Berkovich spaces associated with simple varieties (the projective line, the algebraic curves....).

Then I will try to illustrate the following slogan: 'to see the good analog of a complex object in the p-adic world, one often has to work with Berkovich spaces', through three examples: spectral theory; dynamical systems; and the theory of real (p,q)-forms and related notions (integrals, boundary integrals, curvature forms of metrized line bundles) that as been recently developped in the framework of Berkovich spaces by Chambert-Loir and myself, and that I will try to describe in some detail.

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