| Let L be complete discrete valuation field with perfect residue field k and ring of integers O. A classical theorem of Serre and Tate says that an abelian variety A over L admits a model over O which is an abelian scheme if and only if the inertia subgroup of the absolute Galois group of L acts trivially on the first l-adic étale cohomology group of A, for some l prime number different from the characteristic of k. In this talk I want to discuss a logarithmic version of this result. Namely if the wild inertia subgroup of the absolute Galois group of L acts trivially on the first l-adic étale cohomology group of A, for some l prime number different from the characteristic of k, then A admits a projective, log smooth model over O. During the talk I will recall and give examples about some basic facts on this subject, especially about log smoothness. I am not going to assume familiarity with logarithmic geometry. This is a joint work with A. Smeets. |