The ∂-complex on weighted Bergman spaces on Hermitian manifolds

Haslinger, Friedrich; Son, Duong Ngoc

doi:10.1016/j.jmaa.2020.123994

Cited by 3 publications

(16 citation statements)

References 17 publications

(39 reference statements)

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“…Unlike the cases of Segal-Bargmann space [9] and weighted Bergman space with hyperbolic metric [10], this is not a diagonal operator. Nevertheless we can apply the methods from Theorem 5.4 of [10] to get the following Theorem 4.2. Let h be the half hyperbolic metric on the unit ball B n , α < 0, and ψ(z) = α log(1−|z| 2 ).…”

Section: 24)mentioning

confidence: 98%

“…which, under suitable assumptions, will be a densely defined self-adjoint operator, see [10] and [9], where the classical case of the Segal-Bargmann space with the Euclidean metric is treated.…”

Section: Introductionmentioning

confidence: 99%

“…Here, P h,ψ is the orthogonal projection from L 2 (2,0) (M, h, e −ψ ) onto A 2 (2,0) (M, h, e −ψ ), see [10]. If h is Kähler and ( ∂ψ) ♯ is holomorphic then as in the case of 1-forms,…”

Section: Introductionmentioning

confidence: 99%

“…(4.35)are invariant under the action of 1 . Using a standard result in spectral theory (see Lemma 5.1 of[10] or[3]), we can study the spectrum of 1 by study the spectra of its restrictions onto finite dimensional subspaces A 2 (1,0) (m). If n = 1, then each subspace is one-dimensional.…”

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confidence: 99%

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The $\partial$-Operator and Real Holomorphic Vector Fields

Haslinger¹,

Son²

2020

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Let (M, h) be a Hermitian manifold and ψ a smooth weight function on M . The ∂-complex on weighted Bergman spaces A 2 (p,0) (M, h, e −ψ ) of holomorphic (p, 0)-forms was recently studied in [10] and [9]. It was shown that if h is Kähler and a suitable density condition holds, the ∂-complex exhibits an interesting holomorphicity/duality property when ( ∂ψ) ♯ is holomorphic (i.e., when the real gradient field grad h ψ is a real holomorphic vector field.) For general Hermitian metrics this property does not hold without the holomorphicity of the torsion tensor Tp rs .In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds and their relationship to the ∂-complex on weighted Bergman spaces. For Kähler metrics with multi-radial potential functions on C n we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kähler metrics on complex space forms we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold S 2n−1 × S 1 , and the "half" hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions ψ with real holomorphic gradient fields. They provide a wealth of triples (M, h, e −ψ ) of Hermitian non-Kähler manifolds with weights for which the ∂-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the ∂-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the ∂-equation.

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Section: 24)mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The $\partial$-Operator and Real Holomorphic Vector Fields

Haslinger¹,

Son²

2020

Preprint

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“…see [10] for more details. If, in addition, the metric h is Kählerian one has τ = 0 and thus (1.6) ∂ * u = h j ku j ∂ψ ∂ zk , which means the complex vector field (1.7)…”

Section: Introductionmentioning

confidence: 99%

The $\partial$-operator and real holomorphic vector fields

Haslinger

Son

2022

Pure and Applied Mathematics Quarterly

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Let (M, h) be a Hermitian manifold and ψ a smooth weight function on M . The ∂-complex on weighted Bergman spaces A 2 (p,0) (M, h, e −ψ ) of holomorphic (p, 0)-forms was recently studied in [10] and [9]. It was shown that if h is Kähler and a suitable density condition holds, the ∂-complex exhibits an interesting holomorphicity/duality property when ( ∂ψ) is holomorphic (i.e., when the real gradient field grad h ψ is a real holomorphic vector field.) For general Hermitian metrics, this property does not hold without the holomorphicity of the torsion tensor T p rs . In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds and their relationship to the ∂-complex on weighted Bergman spaces. For Kähler metrics with multi-radial potential functions on C n , we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kähler metrics on complex space forms, we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold S 2n−1 × S 1 and the "half" hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions ψ with real holomorphic gradient fields. They provide a wealth of triples (M, h, e −ψ ) of Hermitian non-Kähler manifolds with weights for which the ∂-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the ∂-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the ∂equation.

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Unbounded Operators on the Segal–Bargmann Space

Haslinger

2024

Springer Proceedings in Mathematics &Amp; Statistics

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The ∂-complex on weighted Bergman spaces on Hermitian manifolds

Cited by 3 publications

References 17 publications

The $\partial$-Operator and Real Holomorphic Vector Fields

The $\partial$-Operator and Real Holomorphic Vector Fields

The $\partial$-operator and real holomorphic vector fields

Unbounded Operators on the Segal–Bargmann Space

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