Semistable reductions and minimalities of invariants for group scheme actions on projective schemes

TL;DR

This paper introduces new invariants and loci related to group actions on projective schemes over non-archimedean fields, establishing their coincidence and non-emptiness under mild conditions, extending prior dynamical results to higher dimensions.

Contribution

It defines the minimal invariant locus and semistable reduction translation locus, proving their coincidence and non-emptiness in higher-dimensional settings under mild assumptions.

Findings

Coincidence of minimal invariant locus and semistable reduction translation locus.

Non-emptiness of these loci under mild completeness assumptions.

Extension of Rumely's 1-dimensional results to higher dimensions.

Abstract

Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\operatorname{SSRL}_x$ in the translation space $\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumption, also their non-emptiness. In the dynamical setting which has been studied by Szpiro--Tepper--Williams and Rumely, the coincidence result is already new in higher dimensions, and the non-emptiness result includes Rumely's $1$-dimensional result at least in the spherical complete case.