Complex-analytic quotients of algebraic $$G$$ G -varieties

Greb, Daniel

doi:10.1007/s00208-014-1163-y

Cited by 4 publications

(2 citation statements)

References 44 publications

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“…Here we recall the basic notions for such quotients. The following definition appears in [HMP98,Gre15] for reduced complex analytic spaces.…”

Section: 3mentioning

confidence: 99%

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Moduli stacks of semistable sheaves and representations of Ext-quivers

Toda

2017

Preprint

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We show that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Extquivers with convergent relations. When the underlying variety is a Calabi-Yau 3-fold, our result describes the above moduli stacks as critical locus analytic locally on the coarse moduli spaces. The results in this paper will be applied to the wall-crossing formula of Gopakumar-Vafa invariants defined by Maulik and the author.

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“…Here we recall the basic notions for such quotients. The following definition appears in [HMP98,Gre15] for reduced complex analytic spaces.…”

Section: 3mentioning

confidence: 99%

“…An analytic Hilbert quotient is known to exist when Z is a reduced Stein space, which is unique up to isomorphism [Hei91]. In [HMP98,Gre15], analytic Hilbert quotients are discussed under the assumption that Z is reduced. It seems that such quotients for non-reduced analytic spaces are not available in literatures.…”

Section: 3mentioning

confidence: 99%

Moduli stacks of semistable sheaves and representations of Ext-quivers

Toda

2017

Preprint

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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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Stratification of singular hyperkähler quotients

Mayrand

2022

Complex Manifolds

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Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

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Complex-analytic quotients of algebraic $$G$$ G -varieties

Cited by 4 publications

References 44 publications

Moduli stacks of semistable sheaves and representations of Ext-quivers

Moduli stacks of semistable sheaves and representations of Ext-quivers

Reductive quotients of klt singularities

Stratification of singular hyperkähler quotients

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