Uniqueness of K-polystable degenerations of Fanovarieties

Harold Blum; Chenyang Xu

doi:10.4007/annals.2019.190.2.4

Uniqueness of K-polystable degenerations of Fanovarieties

Harold Blum, Chenyang Xu

Research output: Contribution to journal › Article › peer-review

65 Scopus citations

Abstract

We prove that K-polystable degenerations of ℚ -Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable 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 K-stable ℚ-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable ℚ-Fano variety is finite.

Original language	English (US)
Pages (from-to)	609-656
Number of pages	48
Journal	Annals of Mathematics
Volume	190
Issue number	2
DOIs	https://doi.org/10.4007/annals.2019.190.2.4
State	Published - 2019
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics (miscellaneous)

Keywords

Degenerations
Fano varieties
K-stability
Moduli

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10.4007/annals.2019.190.2.4

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abstract = "We prove that K-polystable degenerations of ℚ -Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable 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 K-stable ℚ-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable ℚ-Fano variety is finite.",

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year = "2019",

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AU - Blum, Harold

AU - Xu, Chenyang

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N2 - We prove that K-polystable degenerations of ℚ -Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable 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 K-stable ℚ-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable ℚ-Fano variety is finite.

AB - We prove that K-polystable degenerations of ℚ -Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable 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 K-stable ℚ-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable ℚ-Fano variety is finite.

KW - Degenerations

KW - Fano varieties

KW - K-stability

KW - Moduli

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JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -

Uniqueness of K-polystable degenerations of Fanovarieties

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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