# Hodge theory for non-Archimedean analytic spaces

# Hodge theory for non-Archimedean analytic spaces

By Deligne's Hodge theory, the integral cohomology groups H^n(X^h, Z) of the C-analytification of a separated scheme X of finite type over C are provided with a mixed Hodge structure, functorial in X. Given a non-Archimedean field K isomorphic to the field of Laurent power series C((z)), there is a functor X \mapsto X^an_K that takes X to the non-Archimedean K-analytification of X_K = X \otimes_C K. I'll describe a Hodge theory for the full subcategory of the category of K-analytic spaces that consists of the strictly K-analytic spaces with the property that each compact analytic subdomain is isomorphic to an analytic domain in a boundaryless space. It extends Deligne's Hodge theory through the above functor and generalizes previously known complex analytic constructions.