The local obstruction to semi-stable reduction for abelian varieties

TL;DR

This paper reviews the development of Grothendieck's local obstruction groups for semi-stable reduction of abelian varieties, including recent advances in equal characteristic local fields, providing a group-theoretic perspective.

Contribution

It offers an overview of the evolution of the theory of local obstructions, with new insights into the equal characteristic case and a focus on dimension-dependent group characterizations.

Findings

Grothendieck's local obstruction groups are characterized by dimension.

Recent work extends the understanding to equal characteristic local fields.

The paper synthesizes developments since Grothendieck's original definition.

Abstract

Grothendieck defined a group that represents the local obstruction for an abelian variety to have semi-stable reduction. These groups were studied by Silverberg and Zarhin and more recently by the author in order to give a group theoretic characterization of them depending only on the dimension. We give an overview of the developments since Grothendieck's definition with the added novelty of the case of equal characteristic local fields.