For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.
II : G roupes de type multiplicatif, et structure des sch\'emas en groupes g\'en\'eraux , Lecture Notes in Mathematics, vol
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Semistable reductions and minimalities of invariants for group scheme actions on projective schemes
For reductive group scheme actions on projective schemes over complete non-archimedean fields, the minimal invariant locus coincides with the semistable reduction translation locus and is non-empty under mild completeness assumptions.