# Logarithmic good reduction of abelian varieties

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## Abstract

Let *K* be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron–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 Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings–Chai et al.

## Keywords

Line Bundle Abelian Variety Ample Line Bundle Regular Model Abelian Scheme## Notes

### Acknowledgments

We are grateful to Dan Abramovich, Lars Halle, Luc Illusie, Klaus Künnemann, Johannes Nicaise, Martin Olsson, Takeshi Saito and Heer Zhao for useful discussions, and to the referee for a careful reading. We acknowledge the support of the European Research Council’s FP7 programme under ERC Grant Agreements \(\sharp \)306610 (MOTZETA, J. Nicaise) and \(\sharp \)615722 (MOTMELSUM, R. Cluckers). The second-named author is a postdoctoral fellow of FWO Vlaanderen (Research Foundation – Flanders).

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