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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.
@article{10_4007_annals_2019_190_2_4,
author = {Harold Blum and Chenyang Xu},
title = {Uniqueness of $\mathrm {K}$-polystable degenerations of {Fano} varieties},
journal = {Annals of mathematics},
pages = {609--656},
publisher = {mathdoc},
volume = {190},
number = {2},
year = {2019},
doi = {10.4007/annals.2019.190.2.4},
mrnumber = {3997130},
zbl = {07107183},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.4/}
}
TY - JOUR
AU - Harold Blum
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Harold Blum; Chenyang Xu. Uniqueness of $\mathrm {K}$-polystable degenerations of Fano varieties. Annals of mathematics, Tome 190 (2019) no. 2, pp. 609-656. doi: 10.4007/annals.2019.190.2.4
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