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

Pages 609-656 from Volume 190 (2019), Issue 2 by Harold Blum, Chenyang Xu

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.

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Keywords
, , ,
Mathematical Subject Classification
Primary: , ,
DOI
https://doi.org/10.4007/annals.2019.190.2.4
MR
3997130
zbMATH
07107183
Milestones
Received: 18 December 2018
Revised: 8 May 2019
Accepted: 9 May 2019
Published online: 22 August 2019

Authors

Harold Blum

University of Utah, Salt Lake City, UT, USA

Chenyang Xu

Beijing International Center for Mathematical Research, Beijing, China

Current address:

Massachusetts Institute of Technology, Cambridge, MA