Por Mário Edmundo (FCUL e CMAFcIO, Universidade de Lisboa).

Abstract: The interplay between analytic geometry over non-archimedean fields and tropical geometry is a very active area with several applications in fields such as algebraic and arithmetic geometry. Recently, Hrushovski and Loeser introduced a model-theoretic account of the Berkovich's analytification of algebraic varieties: given a variety V over a non-archimedean field K, Hrushovski and Loeser associated to V the space \hat{V}, the stable completion of V, and showed a very deep connection between V and the tropical semi-group Γ_{\infty } where Γ is the value group of K: there is a deformation retraction from \hat{V} to a definable subset of some finite power of Γ_{\infty}. An analogous result was earlier proved by Berkovich for V^{an} under strong algebraic restrictions on the variety V. In this talk we report on the ongoing work (with P. Kovacsics and J. Ye) where we develop a sheaf cohomology theory for the spaces \hat{V}. When the field K is maximally complete of rank one, the spaces \hat{V} and |V^{an}| are naturally homeomorphic and we recover results proved by Berkovich for the cohomology groups.