In a work in progress, I defined integral “etale” cohomology and de Rham cohomology for so called bounded non-Archimedean analytic spaces over the field of formal Laurent power series with complex coefficients. The former are local systems of finitely generated abelian groups on a certain log formal complex analytic space, and the latter are finite free modules over the ring of formal power series provided with a Gauss-Manin connection. Both give rise to vector bundles with connections on that log formal complex analytic space, and the bundles are shown to be isomorphic. Furthermore, if a bounded non-Archimedean analytic space has no boundary, its integral and de Rham cohomology form a so called mixed Hodge structure over the log formal complex analytic space. This structure depends functorially on the non-Archimedean space, and in the case when it comes from a complex algebraic variety over a punctured disc, the associated mixed Hodge structure is an extension of the corresponding classical object.
De Kęstutis Česnavičius
De Takeshi Saito
De Miaofen Chen
