Mathematische Annalen

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

Logarithmic good reduction of abelian varieties

  • Alberto Bellardini
  • Arne SmeetsEmail author


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.


Line Bundle Abelian Variety Ample Line Bundle Regular Model Abelian Scheme 
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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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© 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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