Dolbeault cohomology for almost complex manifolds

Cirici, Joana; Wilson, Scott O.

doi:10.48550/arxiv.1809.01416

Cited by 11 publications

(34 citation statements)

References 26 publications

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“…In [4], Cirici and Wilson introduced a generalization of Dolbeault cohomology on almost complex manifolds. Let (M, J) be an almost complex manifold and H p,q µ = ker µ ∩ A p,q µA p+1,q−2 be the µ-cohomology, which is well defined since µ 2 = 0.…”

Section: Bott-chern Cohomology Of Almost Complex Manifoldsmentioning

confidence: 99%

Bott–Chern harmonic forms on complete Hermitian manifolds

Piovani

Томассини

2019

Int. J. Math.

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Let [Formula: see text] be a Hermitian manifold of complex dimension [Formula: see text]. Assume that the torsion of the Chern connection [Formula: see text] is bounded, and that there exists a [Formula: see text]exhausting function [Formula: see text] such that [Formula: see text] are bounded. We characterize [Formula: see text] Bott–Chern harmonic forms, extending the usual result that holds on compact Hermitian manifolds. Finally, if [Formula: see text] is Kähler complete, [Formula: see text], with [Formula: see text] bounded, and the sectional curvature is bounded, then we get a vanishing theorem for [Formula: see text] Bott–Chern harmonic [Formula: see text]-forms, if [Formula: see text].

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Section: Bott-chern Cohomology Of Almost Complex Manifoldsmentioning

confidence: 99%

Bott–Chern harmonic forms on complete Hermitian manifolds

Piovani

Томассини

2019

Int. J. Math.

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“…In [10] Holt and Zhang answered negatively to this question, showing with an explicit example that there exist almost complex structures on the Kodaira-Thurston manifold with Hodge number h 0,1 ∂ varying with different choices of Hermitian metrics. They also proved that if (M, J, g, ω) is a 4-dimensional compact almost-Kähler manifold, then h 1,1 ∂ = b − + 1, where b − denotes the dimension of the space of anti However, as noticed in [5], in special bi-degrees, e.g., (p, 0), the almost-complex Dolbeault cohomology groups have finite dimensions. For this reason, we compute such groups in bi-degree (p, 0), for the families of almost-complex manifolds considered above.…”

Section: Introductionmentioning

confidence: 92%

Almost-complex invariants of families of six-dimensional solvmanifolds

Tardini¹,

Томассини²

2021

Preprint

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We compute almost-complex invariants h p,0 ∂ , h p,0 Dol and almost-Hermitian invariants h p,0 δ on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a symplectic structure on a given compact almost-complex manifold. Notice that, when (X, J, g, ω) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h p,q ∂ .

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“…In [4], Cirici and Wilson introduced a generalization of Dolbeault cohomology on almost complex manifolds. Let (M, J ) be an almost complex manifold and…”

Section: Bott-chern Cohomology Of Almost Complex Manifoldsmentioning

confidence: 99%

Bott–Chern Laplacian on almost Hermitian manifolds

Piovani

Томассини

2022

Math. Z.

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Let $$(M,J,g,\omega )$$ ( M , J , g , ω ) be a 2n-dimensional almost Hermitian manifold. We extend the definition of the Bott–Chern Laplacian on $$(M,J,g,\omega )$$ ( M , J , g , ω ) , proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott–Chern Laplacian coincide. We show that such a property does not hold when $$(M,J,g,\omega )$$ ( M , J , g , ω ) is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira–Thurston manifold. Furthermore, if $$(M,J,g,\omega )$$ ( M , J , g , ω ) is a connected compact almost Hermitian 4-manifold, denoting by $$h^{1,1}_{BC}$$ h BC 1 , 1 the dimension of the space of Bott–Chern harmonic (1, 1)-forms, we prove that either $$h^{1,1}_{BC}=b^-$$ h BC 1 , 1 = b - or $$h^{1,1}_{BC}=b^-+1$$ h BC 1 , 1 = b - + 1 . In particular, if g is almost Kähler, then $$h^{1,1}_{BC}=b^-+1$$ h BC 1 , 1 = b - + 1 , extending the result by Holt and Zhang (Harmonic forms on the Kodaira–Thurston manifold. arXiv:2001.10962, 2020) for the kernel of Dolbeault Laplacian. We also show that the dimensions of the spaces of Bott–Chern and Dolbeault harmonic (1, 1)-forms behave differently on almost complex 4-manifolds endowed with strictly locally conformally almost Kähler metrics. Finally, we relate some spaces of Bott-Chern harmonic forms to the Bott–Chern cohomology groups for almost complex manifolds, recently introduced in Coelho et al. (Maximally non-integrable almost complex structures: an h-principle and cohomological properties, arXiv:2105.12113, 2021).

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Dolbeault cohomology for almost complex manifolds

Cited by 11 publications

References 26 publications

Bott–Chern harmonic forms on complete Hermitian manifolds

Bott–Chern harmonic forms on complete Hermitian manifolds

Almost-complex invariants of families of six-dimensional solvmanifolds

Bott–Chern Laplacian on almost Hermitian manifolds

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