# 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.

## Authors

Harold Blum

University of Utah, Salt Lake City, UT, USA

Chenyang Xu

Beijing International Center for Mathematical Research, Beijing, China