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arXiv · 1512.02464

Logarithmic good reduction of abelian varieties

Alberto Bellardini·Arne Smeets
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Abstract

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.

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Alberto Bellardini, Arne Smeets. 2015-12-08. Logarithmic good reduction of abelian varieties. https://arxiv.org/abs/1512.02464

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