Some Observations on the Dimension of Fano K-Moduli

Martinez-Garcia, Jesus; Spotti, Cristiano

doi:10.1007/978-3-031-17859-7_30

Springer Proceedings in Mathematics &Amp; Statistics

2023

DOI: 10.1007/978-3-031-17859-7_30

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Some Observations on the Dimension of Fano K-Moduli

Jesus Martinez-Garcia

Cristiano Spotti

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2024

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Reductive quotients of klt singularities

Braun,

Greb,

Langlois

et al. 2024

Invent. math.

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We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety $X$ X endowed with the action of a reductive group $G$ G and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$ X → X / / G , we can find a boundary $B$ B on $X/\!/G$ X / / G so that the pair $(X/\!/G,B)$ ( X / / G , B ) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$ G -manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$ n -dimensional K-polystable smooth Fano varieties of volume $v$ v has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

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Reductive quotients of klt singularities

Braun,

Greb,

Langlois

et al. 2024

Invent. math.

View full text Add to dashboard Cite

show abstract

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Some Observations on the Dimension of Fano K-Moduli

Cited by 1 publication

References 12 publications

Reductive quotients of klt singularities

Reductive quotients of klt singularities

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