Kähler-Einstein metrics: Old and New

Angella, Daniele; Spotti, Cristiano

doi:10.1515/coma-2017-0014

Cited by 3 publications

(4 citation statements)

References 131 publications

(128 reference statements)

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“…Given a polarized variety (X, L) or a vector bundle E on (X, L), one considers various stability notions for X and E (see e.g. [1,5] for K-stability of varieties, and [10] for slope stability of bundles). In the presence of symmetries for (X, L), that is, given an algebraic action of a reductive Lie group G on (X, L), it is natural to ask whether these stability notions persist on the GIT quotient Y of (X, L) by G. By the Yau-Tian-Donaldson conjecture [1,5] and the Kobayashi-Hitchin correspondence [13], the stability of (X, L) or E can be related to the existence of a canonical metric on the underlying complex object, variety or bundle.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, there is at most one class α on Y satisfying (i) up to scale. 1 Also known as toric reflexive sheaves in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…Given a polarized variety or a vector bundle on , one considers various stability notions for and (see e.g. [1, 5] for K-stability of varieties, and [10] for slope stability of bundles). In the presence of symmetries for , that is, given an algebraic action of a reductive Lie group on , it is natural to ask whether these stability notions persist on the GIT quotient of by .…”

Section: Introductionmentioning

confidence: 99%

“…In the presence of symmetries for , that is, given an algebraic action of a reductive Lie group on , it is natural to ask whether these stability notions persist on the GIT quotient of by . By the Yau-Tian-Donaldson conjecture [1, 5] and the Kobayashi-Hitchin correspondence [13], the stability of or can be related to the existence of a canonical metric on the underlying complex object, variety or bundle. From this differential geometrical point of view, -orbits can detect curvature on , and canonical metrics are not necessarily preserved under GIT quotients.…”

Section: Introductionmentioning

confidence: 99%

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Equivariant stable sheaves and toric GIT

Clarke¹,

Tipler²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$ . We show, under a genericity assumption on $G$ , that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$ , we relate stable equivariant reflexive sheaves on certain $(n-1)$ -dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$ .

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Section: Introductionmentioning

confidence: 99%

“…Moreover, there is at most one class α on Y satisfying (i) up to scale. 1 Also known as toric reflexive sheaves in the literature.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivariant stable sheaves and toric GIT

Clarke¹,

Tipler²

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Constant $$\mu $$-Scalar Curvature Kähler Metric—Formulation and Foundational Results

Inoue

2022

J Geom Anal

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Constant $μ$-scalar curvature Kähler metric -- formulation and foundational results

Inoue¹

2019

Preprint

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We introduce µ-scalar curvature for a Kähler metric with a moment map µ and start up a study on constant µ-scalar curvature Kähler metric as a generalization of both cscK metric and Kähler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant µ-scalar curvature Kähler metric by investigating a volume functional as a generalization of Tian-Zhu's work, which is closely related to Perelman's W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant µ-scalar curvature and as a prelude to new K-stability concept. Contents 18 4. µK-energy and µK-stability 25 4.1. µK-energy functional 25 4.2. A prelude to µK-stability 29 5. Openness of Kähler class and λ admitting µ-cscK 30 5.1. Regularity 30 5.2. Perturbation 30 References 32

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Kähler-Einstein metrics: Old and New

Cited by 3 publications

References 131 publications

Equivariant stable sheaves and toric GIT

Equivariant stable sheaves and toric GIT

Constant $$\mu $$-Scalar Curvature Kähler Metric—Formulation and Foundational Results

Constant $μ$-scalar curvature Kähler metric -- formulation and foundational results

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