Logarithmic good reduction of abelian varieties

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 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.

Publication year:2017