Reductive quotients of klt singularities
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866917752717967360 |
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| author | Braun, Lukas Greb, Daniel Langlois, Kevin Moraga, Joaquín |
| author_facet | Braun, Lukas Greb, Daniel Langlois, Kevin Moraga, Joaquín |
| contents | We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ endowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ 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 $G$-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 $n$-dimensional K-polystable Fano manifolds of volume $v$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_02812 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Reductive quotients of klt singularities Braun, Lukas Greb, Daniel Langlois, Kevin Moraga, Joaquín Algebraic Geometry Complex Variables Differential Geometry Primary 14B05, 14E30, 14L24, 14M25, Secondary 14A20, 53D20 We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ endowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ 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 $G$-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 $n$-dimensional K-polystable Fano manifolds of volume $v$ 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. |
| title | Reductive quotients of klt singularities |
| topic | Algebraic Geometry Complex Variables Differential Geometry Primary 14B05, 14E30, 14L24, 14M25, Secondary 14A20, 53D20 |
| url | https://arxiv.org/abs/2111.02812 |