Stability under deformations of extremal almost-Kähler metrics in dimension $4.$

Lejmi, Mehdi

doi:10.4310/mrl.2010.v17.n4.a2

Cited by 20 publications

(22 citation statements)

References 23 publications

(37 reference statements)

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“…where ∆ d is the Riemannian Laplacian with respect to the metric g(•, •) = ω(•, J•) (here we use the convention g(ω, ω) = n). By studying Lejmi's operator P J [9], Tan-Wang-Zhou [13] proved that J is C ∞ -pure and full when dim(ker…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…If the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms, i.e., H 2 dR (X) ⊂ H 2 d+d Λ (X), then identity b 2 ≤ h 2 d+d Λ naturally holds. In the fifth section of [9], Lejmi introduced the differential operator P J on a closed almost Kähler 4-manifold (X, J) Tan-Wang-Zhou [13] extended the defined to higher dimensions. We can give a sufficient condition for H 2 dR (X) ⊂ H 2 d+d Λ (X).…”

Section: Introductionmentioning

confidence: 99%

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On second non-HLC degree of closed symplectic manifold

Huang

2021

Preprint

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In this note, we show that for a closed almost-Kähler manifold (X, J) with the almost complex structure J satisfies dim ker P J = b 2 − 1 the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. In particular, suppose that X is four-dimension, if the self-dual Betti number b + 2 = 1, then we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms.

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Section: Proof Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On second non-HLC degree of closed symplectic manifold

Huang

2021

Preprint

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“…None of these problems reduces to the previously studied problem of finding extremal almost Kähler structures, that by definition are the critical points of the squared norm of the Hermitian scalar curvature on J(M, Ω). That problem is surveyed in the last section of [2] and studied in [47,46]. There are potentially interesting connections between these various problems, but elucidating them is not the purpose of this article.…”

Section: 2mentioning

confidence: 99%

Critical symplectic connections on surfaces

Fox

2019

Journal of Symplectic Geometry

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The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal Kähler metrics. In particular, moment constant connections are critical, where a symplectic connection is critical if it is critical, with respect to arbitrary variations, for the L 2 -norm of the Cahen-Gutt moment map. This occurs if and only if the Hamiltonian vector field generated by its moment map image is an infinitesimal automorphism of the symplectic connection. This paper develops the study of moment constant and critical symplectic connections, following, to the extent possible, the analogy with the similar, but different, setting of constant scalar curvature and extremal Kähler metrics.It focuses on the special context of critical symplectic connections on surfaces, for which general structural results are obtained, although some results about the higher-dimensional case are included as well. For surfaces, projectively flat and preferred symplectic connections are critical, and the relations between these and other related conditions are examined in detail. The relation between the Cahen-Gutt moment map and the Goldman moment map for projective structures is explained.

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“…We choose m = e sin 2π(x 2 +x 4 ) to be a positive periodic function on J is C ∞ pure and full.). Additionally, Lejmi [16] proved that, on a closed almost Kähler 4-manifold (M, g, J, ω), the kernel of P J consists of primitive harmonic 2forms and dim ker…”

Section: So We Will Getmentioning

confidence: 99%

Primitive cohomology of real degree two on compact symplectic manifolds

2015

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In this paper, we define the generalized Lejmi's P J operator on a compact almost Kähler 2n-manifold. We get that J is C ∞ -pure and full if dim ker P J = b 2 − 1. Additionally, we investigate the relationship between J-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic 4-manifold. (2000): 53C55, 53C22. AMS Classification

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Stability under deformations of extremal almost-Kähler metrics in dimension $4.$

Cited by 20 publications

References 23 publications

On second non-HLC degree of closed symplectic manifold

On second non-HLC degree of closed symplectic manifold

Critical symplectic connections on surfaces

Primitive cohomology of real degree two on compact symplectic manifolds

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