Toric generalized Kähler–Ricci solitons with Hamiltonian 2-form

Legendre, Eveline; Tønnesen-Friedman, Christina W.

doi:10.1007/s00209-012-1112-y

Cited by 9 publications

(5 citation statements)

References 27 publications

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“…Apostolov et al [2,3,4,5] use c-projectively equivalent metrics (in the guise of Hamiltonian 2-forms) to study extremal Kähler metrics, where the scalar curvature of a Kähler metric lies in the kernel of its c-projective Hessian, and it would be natural to consider extremal quasi-Kähler metrics in the same light. Kähler-Ricci solitons (and generalisations) admitting c-projectively equivalent metrics have also been studied in special cases [5,67,71], but the picture is far from complete.…”

Section: Global Resultsmentioning

confidence: 99%

“…Of course, as soon as one speaks of holomorphic functions on a complex manifold one is obliged to work with complexvalued differential forms. However, even if one is concerned only with real-valued forms and tensors, it is convenient firstly to decompose the complex versions and then impose reality as, for example, in (67). In fact, this is already a feature of representation theory in general.…”

Section: Bgg Sequencesmentioning

confidence: 99%

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C-projective geometry

Calderbank¹,

Eastwood²,

Matveev³

et al. 2020

Memoirs of the AMS

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We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kähler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kähler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata Conjecture for complete Kähler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric. Proof. By assumption, we haveIf ∇ a q = 0, it follows easily that p and q are locally constant hence constant. Otherwise, contracting this expression with a nonzero tangent vector X a in the kernel of ∇ a q, we deduce that Λ b = qΛ b and ∇ a p = −ξ∇ a q for some function ξ., it follows from what we have already proven that ξ is constant. Otherwise, we deduce that p and q are constant.

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Section: Global Resultsmentioning

confidence: 99%

Section: Bgg Sequencesmentioning

confidence: 99%

C-projective geometry

Calderbank¹,

Eastwood²,

Matveev³

et al. 2020

Memoirs of the AMS

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“…In the toric setting, (W, ω) is monotone if and only the associated labelled polytope (P, l) is monotone in the sense of Definition 3.2. The proof of this fact works for non-Delzant labelled polytope, see eg [14] and [26,Lemma 2.4], as it amounts to compare the Ricci potential and the Kähler potential as functions on the moment polytope. That is, to check if there exist p ∈ t * , λ > 0 such that…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Toric Sasaki-Einstein metrics with conical singularities

Borbon¹,

Legendre²

2020

Preprint

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We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

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“…Note that by Proposition 2.2 of [CFO08] the Lie algebra of holomorphic Hamiltonian vector fields defined in [FOW09] coincides with the Lie algebra of transverse holomorphic vector fields. We mention also that Sasaki-Ricci solitons on toric 5-manifolds were studied in [LTF13]. Definition 6.16.…”

Section: Now By Equationmentioning

confidence: 99%

The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature

Boyer

Tønnesen-Friedman

2015

J Geom Anal

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We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki-Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension 2n + 1 and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds M l1,l2,w of dimension 2n + 3 which depend on four integer parameters l 1 , l 2 , w 1 , w 2 . Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two dimensional subcone of Sasaki Ricci solitons in the Sasaki cone, and a unique Sasaki-Einstein metric in each of the two dimensional sub cones.2000 Mathematics Subject Classification. Primary: 53D42; Secondary: 53C25.

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Toric generalized Kähler–Ricci solitons with Hamiltonian 2-form

Cited by 9 publications

References 27 publications

C-projective geometry

C-projective geometry

Toric Sasaki-Einstein metrics with conical singularities

The Sasaki Join, Hamiltonian 2-Forms, and Constant Scalar Curvature

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