Semi-abelian reduction of Albanese varieties

Abstract

Let X_K be a projective geometrically normal variety over the fraction field of a DVR. We show that if X_K admits a projective model whose special fibre has at worst nodes in codimension one, then the Albanese variety Alb_{X_K/K} has semi-abelian reduction over the same base. As an application, we extend the classical nonexistence theorem of Fontaine and Abrashkin for abelian schemes over Z to the singular setting. For instance, if a flat projective Z-scheme X has normal and geometrically connected fibres, then H^1(X_Q, O_{X_Q})=0.