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Mathematische Annalen

, Volume 369, Issue 3–4, pp 1435–1442 | Cite as

Logarithmic good reduction of abelian varieties

  • Alberto Bellardini
  • Arne SmeetsEmail author
Article
  • 443 Downloads

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department WiskundeUniversity of LeuvenHeverleeBelgium
  2. 2.Radboud Universiteit Nijmegen, IMAPPNijmegenThe Netherlands

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