Let $K$ be a discrete valuation field with perfect residue field $k$ and ring of integers $\mathcal{O}_K$. A classical theorem of Serre and Tate says that an abelian variety $A$ over $K$ admits a model over $\mathcal{O}_K$ which is an abelian scheme if and only if the inertia group $I_K\subset Gal(\overline{K}/K)$ acts trivially on $H_{et}^1(A_{\overline{K}},\mathbb{Q}_l)$ for $l$ a prime number different from $char(k)$.

In this talk I want to discuss a logarithmic version of this result. Namely, let $A$ be an abelian variety over $K$ and assume that the dimension of $A$ is less than 5. If the wild inertia $P_K\subset I_K$ acts trivially on $H_{et}^1(A_{\overline{K}},\mathbb{Q}_l)$ for some prime $l\neq char(k)$ then $A$ admits a projective, log-smooth model over $\mathcal{O}_K$.

I am not going to assume familiarity with logarithmic geometry. During the talk, I will recall and give examples about some basic facts on this subject, especially about logarithmic smoothness.

This is a joint work with A. Smeets.