Projectivity of analytic Hilbert and Kähler quotients

Greb, Daniel

doi:10.1090/s0002-9947-10-05000-2

Cited by 12 publications

(25 citation statements)

References 38 publications

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“…217,after Main Theorem]. Note however that algebraicity phenomena similar to the one observed here have been discovered earlier in Kähler Reduction Theory and Geometric Invariant Theory, see for example [HM01] or [Gre10].…”

Section: Introductionsupporting

confidence: 78%

Variation of Gieseker moduli spaces via quiver GIT

2016

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We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in N^1(X)_\mathbb{R}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus-flips.Comment: 51 pages; v2: added discussion concerning characteristic of the base field, fixed some typos and inconsistencies; removed "I" from title as second part has a new title; to appear in Geometry & Topolog

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Section: Introductionsupporting

confidence: 78%

Variation of Gieseker moduli spaces via quiver GIT

2016

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“…Since H is reductive, there exists a Zariski-locally closed H-invariant subvariety S of U such that the natural map ϕ : G × H S → U is locally biholomorphic at the point [e, p] ∈ G × H S, cf. the proof of Theorem 4.6 in [Gre08a]. Furthermore, shrinking S, we may assume that S is affine, and that ϕ(G × H S) is π-saturated in U.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Proof. Since we are only interested in the behaviour of π over a big open set of Q, by Corollary 9.3 we may assume that there exist an irreducible variety B, an affine algebraic G-variety F with C[F] G = C, and an étale G-equivariant surjective map ψ : B × F → U, such that the induced mapψ : B → Q is étale, and such that in the commutative diagram [Gre08a], we conclude that ψ|ψ−1 (W)×F :ψ −1 (W) × F → π −1 (W) is likewise proper and therefore finite.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…For a homogeneous section f ∈ A we define the zero set to be Z( f ) := supp div( f ) + D . Given a G-linearised Weil divisor D as above, a point x ∈ X is called semistable with respect to D if there exists an integer n > 0 and a G- [Gre08a]. A momentum map with respect to ω is a K-equivariant smooth map µ : X → k * such that…”

Section: Analytic Hilbert Quotients and Geometric Invariant Theorymentioning

confidence: 99%

“…Note that we cannot apply Theorem 6.1 to X(µ), as it is a priori not evident that X(µ) is an analytically Zariski-open subset of X. However, the proof of part (1) in [Gre08a] is quite similar to the proof of Theorem 6.1 in Section 6.2.…”

Section: Kähler Quotients Of Algebraic Hamiltonian G-varietiesmentioning

confidence: 99%

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Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Greb

2010

Advances in Mathematics

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Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient.Mathematical Subject Classification: 14L30, 14L24, 32M05, 53D20, 53C55.

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

Greb

2015

Math. Ann.

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Abstract. It is shown that any compact semistable quotient of a normal variety by a complex reductive Lie group G (in the sense of Heinzner and Snow) is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class QG has a realisation as a good quotient, and that every complete algebraic variety in QG is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class QT , where T is an algebraic torus, is a toric variety.

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Projectivity of analytic Hilbert and Kähler quotients

Cited by 12 publications

References 38 publications

Variation of Gieseker moduli spaces via quiver GIT

Variation of Gieseker moduli spaces via quiver GIT

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

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

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