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

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 ${\mathrm{MinInvLoc}}_x$ and the semistable reduction translation locus ${\mathrm{SSRL}}_x$ in the translation space ${\mathrm{BT}}_G(K)$ 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.