Uniqueness of $\mathrm {K}$-polystable degenerations of Fano varieties

Abstract

We prove that $\mathrm {K}$-polystable degenerations of $\mathbb {Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of $\mathrm {K}$-stable $\mathbb {Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly $\mathrm {K}$-stable $\mathbb {Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a $\mathrm {K}$-stable $\mathbb {Q}$-Fano variety is finite.