Joint IAS/PU Arithmetic Geometry

The relationship between $p$-adic (pro)-etale cohomology and de Rham cohomologies is well understood now for algebraic varieties over local fields of mixed characteristic. For analytic varieties it remains largely conjectural though much progress has been made in the last decade especially in the Stein case. I will survey the results as well as the conjectures. This is based on a joint work with Pierre Colmez and Sally Gilles.