Generalized Ricci Flow

Garcia-Fernandez, Mario; Streets, Jeffrey

doi:10.48550/arxiv.2008.07004

Cited by 2 publications

(5 citation statements)

References 111 publications

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“…The Calabi-Yau type problem of generalized Kähler manifolds of type (0, 0) was discussed by Apostolov and Streets in [2]. A generalized Kähler Ricci flow has been explored [7], [3]. It is a remarkable problem to investigate a relation between the scalar curvature as moment map in this paper and the generalized Kähler Ricci flow.…”

Section: Scalar Curvature Of Generalized Kähler Manifoldsmentioning

confidence: 94%

Scalar curvature as moment map in generalized Kähler geometry

Goto

2020

Journal of Symplectic Geometry

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It is known that the scalar curvature arises as the moment map in Kähler geometry. In pursuit of the analogy, we develop the moment map framework in generalized Kähler geometry of symplectic type. Then we establish the definition of the scalar curvature on a generalized Kähler manifold of symplectic type from the moment map view point. We also obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. We show that infinitesimal deformations of generalized Kähler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kähler manifold of type (0, 0). Poisson structures constructed from a Kähler action of T m on a Kähler-Einstein manifold give rise to intriguing deformations of generalized Kähler-Einstein structures. In particular, the anticanonical divisor of three lines on CP 2 in general position yields nontrivial examples of generalized Kähler-Einstein structures. IntroductionLet (X, ω) be a compact symplectic manifold with a symplectic structure ω. An almost complex structure J is compatible with ω if a pair (J, ω) gives an almost Kähler structure on M . We denote byC ω the set of almost complex structures which are compatible with ω. ThenC ω is an infinite dimensional Kähler manifold on which Hamiltonian diffeomorphisms of (M, ω) actC ω preserving the Kähler structure. Each J ∈ C ω gives a Riemannian metric g(J) and we denote by s(J) the scalar curvature of g(J) which is regarded as a function on C ω . Then the following theorem was established in Kähler geometry by Fujiki and Donaldson.The scalar curvature is the moment map onC ω for the action of Hamiltonian diffeomorphisms.The moment map framework in Kähler geometry suggests that the existence of constant scalar curvature Kähler metrics is inevitably linked with the certain stability in algebraic geometry which leads to wellknown Donaldson-Tian-Yau conjecture in Kähler geometry.Generalized Kähler geometry is a successful generalization of ordinary Kähler geometry which is equivalent to bihermitian geometry satisfying the certain torsion conditions. Many interesting examples of generalized Kähler manifolds were already constructed by holomorphic Poisson structures [Go1], [Go2], [Go3], [Go4], [Gu1], [Hi1], [Hi2], [Lin1]. * 1 Thus ψ is given by ψ = e b+ √ −1ω , where ω is a real symplectic form. A pair (J , J ψ ) is called a generalizedKähler structure of symplectic type. We can obtain further generalization of moment map framework for any d-closed nondegenerate, pure spinor ψ. J . Since h 0,2 · φ α = 0, we haveThus we have e · φ α , N · h · φ α s = 0. Then it follows N · h · φ α , e · φ α s = 0.Lemma 7.10. Let N be as in before. Then we haveProof. Let {e i } be a local basis of E J . Since dφ α = η α · φ α + N · φ α , then we have N · φ α , e i · e j · e k · φ α s = dφ α , e i · e j · e k · φ α s = − e i · e j · dφ α , e k · φ α s Since the co...

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Section: Scalar Curvature Of Generalized Kähler Manifoldsmentioning

confidence: 94%

Scalar curvature as moment map in generalized Kähler geometry

Goto

2020

Journal of Symplectic Geometry

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“…Killing spinors on exact Courant algebroids. In this section we study the Killing spinor equations on an exact Courant algebroid [22,24], for Riemannian generalized metrics and a special class of divergences called closed [28]. As we will see in Proposition 4.5, restricting to invariant solutions on a homogeneous manifold, we obtain Killing spinors on a quadratic Lie algebra in the sense of Definition 2.13.…”

mentioning

confidence: 99%

“…where µ g is the volume element of g. Following [28], we introduce next a compatibility condition for pairs (V + , div).…”

mentioning

confidence: 99%

“…Given a compatible pair (V + , div), we have that e = X + ϕ in the splitting determined by V + and the infinitesimal isometry condition reads (see [28,Lemma 2.50])…”

mentioning

confidence: 99%

“…where F i = Ann(∧ i+1 t). Given now T-dual pairs (M, τ ) and ( M , τ ), there exists representatives H ∈ τ and Ĥ ∈ τ in F 1 of M and M , respectively, which satisfy the conditions in Definition 4.10 (see [28,Lemma 10.5]).…”

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(0,2) Mirror Symmetry on homogeneous Hopf surfaces

Álvarez-Cónsul¹,

Hera²,

Garcia-Fernandez³

2020

Preprint

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In this work we find the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. For this we follow Borisov's approach to mirror symmetry using vertex algebras and the chiral de Rham complex. Our examples of (0,2) mirrors are given by pairs of Hopf surfaces endowed with a Bismut-flat pluriclosed metric. Requiring that the geometry is homogeneous, we reduce the problem to the study of Killing spinors on a quadratic Lie algebra and the construction of associated N = 2 superconformal structures on the superaffine vertex algebra, combined with topological T-duality.

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Generalized Ricci Flow

Cited by 2 publications

References 111 publications

Scalar curvature as moment map in generalized Kähler geometry

Scalar curvature as moment map in generalized Kähler geometry

(0,2) Mirror Symmetry on homogeneous Hopf surfaces

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