  Let $K$ be a field which is complete for a discrete valuation. We prove a
logarithmic version of the N\'eron-Ogg-Shafarevich criterion: if $A$ is an
abelian variety over $K$ which is cohomologically tame, then $A$ has good
reduction in the logarithmic setting, i.e. there exists a projective, log
smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the
existence of a projective, regular model of $A$, generalizing a result of
K\"unnemann. The proof combines a deep theorem of Gabber with the theory of
degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.
