Sean Howe (University of Utah)

Abstract: In complex geometry, it is an extremely useful fact that universal covers of complex manifolds have tangent spaces and that period maps arising from the cohomology of families of varieties are differentiable. In p-adic geometry, unfortunately, it is known that the covers of rigid analytic varieties trivializing local systems of etale cohomology do not admit a good theory of Kahler differentials. In this talk, we explain how, building from first principles and a single clever idea, one can nonetheless assign tangent spaces to many of the perfectoid spaces and diamonds that arise naturally in the study of rigid analytic varieties and their cohomology and then differentiate period maps. These spaces provide, in particular, a natural conceptual framework for predicting when a diamond is a perfectoid space. In this talk we will focus mostly on examples in the theory of local Shimura varieties and explain the relation to work of Johannson, Ludwig and Hansen on perfectoid quotients of Lubin-Tate space, Ivanov and Weinstein on cohomological smoothness, and Pan and Camargo on geometric Sen theory and sheaves of locally analytic functions. This is joint work with Peter Wear.