Let $K$ be a field complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some prime $\ell \neq p$) the action of the absolute Galois group of $K$ on the $\ell$-adic Tate module $T\_\ell(A)$ is tamely ramified.


One could hope that under this condition, there is a projective model for $A$ over $\mathcal{O}\_K$ which is *log smooth*; here the log structure is the one induced by the special fibre. I am not sure whether this is known; does anyone know a reference for this?


If one can find a projective, semistable, Galois-equivariant model for the abelian variety after a tamely ramified base change, then I know how to go from there - but my argument uses heavy machinery... There is a nice paper by Künnemann (Duke 1998) in which he proves that one can find a projective semistable model after a finite base change. But it is not clear to me whether a tame base change is sufficient to obtain such a model if one assumes moreover that the Galois action on the Tate module is tamely ramified.



Log smooth models for abelian varieties
This question has been answered in the following preprint:


<http://arxiv.org/abs/1512.02464>


