Description:

In the study of families of complex varieties, the variation of the Hodge filtration on singular cohomology plays an important role. This variation can be encoded in the Hodge period map, whose derivative is computed from the Kodaira-Spencer class of a deformation. Similarly, in p-adic geometry, the variation of the Hodge-Tate filtration on p-adic etale cohomology provides an important tool for studying families of rigid analytic varieties. This variation is encoded in the Hodge-Tate period map (or better, its lattice refinement), but until recently this map did not exist within a theory where it made sense to talk about derivatives. In this talk, we explain how to differentiate lattice Hodge-Tate period maps using the theory of inscribed v-sheaves; along the way, we give concrete examples and explain connections with classical notions like the Kodaira-Spencer class as well as other recent developments in p-adic Hodge theory and p-adic geometry.