Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Poon, Yat Sun; Simanyi, John

doi:10.1515/coma-2019-0004

Cited by 6 publications

(8 citation statements)

References 21 publications

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“…It generalizes a computation on Kodaira surface in [46] from complex 2-dimension to all complex dimensions. It also generalizes an observation in [49] that the holomorphic Poisson cohomology of any holomorphic Poisson structures on primary Kodaira manifolds of all dimensions, as Gerstenhaber algebra, are isomorphic to H • (M).…”

Section: Deformation Theorysupporting

confidence: 67%

“…Based on such foundation, the author and his collaborators recently identify when the spectral sequence degenerates on the first page if the complex structure is abelian and the underlying nilmanifold is 2-steps [48] [49]. It enables an effective computation of holomorphic Poisson cohomology.…”

Section: While Classical Complex Deformation Has Been a Century-old S...mentioning

confidence: 99%

“…To make a computation effective, one may consider an ascending basis for the algebra g 1,0 and the conjugation of its dual for g 0,1 as in [49]. Precisely, given the work of [13] [14], Salamon finds [53] that there exists a basis {ω 1 , .…”

Section: Further Developmentmentioning

confidence: 99%

“…In [49], the author and his collaborator further identify conditions to secure an isomorphism between Gerstenbaher algebras:…”

Section: Further Developmentmentioning

confidence: 99%

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Frobenius Structures and Generalized Deformation of Kodaira Manifolds

Poon¹

2021

Preprint

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It is known that generalized deformation in the sense of Hitchin-Gaultieri is a geometric realization of the degree-2 component of Kontsevich-Barannikov's homological approach to extended deformation. Through extended deformation, one associates a Frobenius structure to the extended moduli space. In this notes, we prove that on primary Kodaira manifolds the restriction of the Frobenius structure on the degree-2 component of the extended moduli space is trivial. It generalizes the author's past observation on Kodaira surface.

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Section: Deformation Theorysupporting

confidence: 67%

Section: While Classical Complex Deformation Has Been a Century-old S...mentioning

confidence: 99%

Section: Further Developmentmentioning

confidence: 99%

“…In [49], the author and his collaborator further identify conditions to secure an isomorphism between Gerstenbaher algebras:…”

Section: Further Developmentmentioning

confidence: 99%

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Frobenius Structures and Generalized Deformation of Kodaira Manifolds

Poon¹

2021

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“…where the differential operator b π (−) = [π, −] is the adjoint action of π with respect to the Schouten bracket; see [25,23]. This cohomology has been widely studied; see, for example, [20,12,10,7,27,28,19] and references therein. Dually, from a homological point of view, we have the so called holomorphic Koszul-Brylinski complex:…”

Section: Introductionmentioning

confidence: 99%

Holomorphic Koszul-Brylinski homologies of Poisson blow-ups

Chen¹,

Chen²,

Yang³

et al. 2022

Preprint

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We derive a blow-up formula for holomorphic Koszul-Brylinski homologies of compact holomorphic Poisson manifolds. As applications, we investigate the invariance of the E1-degeneracy of the Dolbeault-Koszul-Brylinski spectral sequence under Poisson blowups, and compute the holomorphic Koszul-Brylinski homology for del Pezzo surfaces and two complex nilmanifolds with holomorphic Poisson structures.Question. For a non-trivial holomorphic Poisson structure, can we describe explicitly the variance of the holomorphic Koszul-Brylinski homology under a Poisson blow-up?Using a sheaf-theoretic approach, we establish a blow-up formula for holomorphic Koszul-Brylinski homology as follows.Theorem 1.1. Suppose (X, π) is a compact holomorphic Poisson manifold of complex dimension n ≥ 2, and (Z, π| Z ) ⊂ (X, π) is a closed holomorphic Poisson submanifold of codimension c ≥ 2 with trivial transverse Poisson structure. Let ϕ : X → X be the blow-up of X along Z and π be the unique holomorphic Poisson structure on X such that ϕ is a Poisson morphism, i.e., ϕ ⋆ π = π. Then there exists an isomorphism of holomorphic Koszul-Brylinski homologies:for any 0 ≤ k ≤ 2n. In particular, there exists an isomorphismObserve that the first page of the Dolbeault-Koszul-Brylinski spectral sequence of the holomorphic Poisson manifold (X, π) is the Dolbeault cohomology:The study of the degeneracy of the Dolbeault-Koszul-Brylinski spectral sequence at E 1 -page may be of independent interest. As an application of Theorem 1.1, we investigate the invariance of such degeneracy under Poisson blow-ups.Theorem 1.2. With the assumption of Theorem 1.1, the Dolbeault-Koszul-Brylinski spectral sequence for ( X, π) degenerates at E 1 -page if and only if it does so for (X, π) and (Z, π| Z ).This paper is organized as follows. In §2, we review some basics on holomorphic Poisson manifolds and the holomorphic Koszul-Brylinski homology. We devote §3 to Poisson blow-ups and modifications. In §4 we derive the Poisson projective bundles formula for holomorphic Koszul-Brylinski homology, a key part of the proof of the main theorems. In §5, the proofs of the main theorems are given. In §6, the holomorphic Koszul-Brylinski homologies of some compact holomorphic Poisson manifolds are computed. Finally, the Appendix A gives the Hodge diamond of a six-dimensional complex nilmanifold in § §6.3.

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Holomorphic Koszul–Brylinski homology via Dolbeault cohomology

Meng

2022

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Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Cited by 6 publications

References 21 publications

Frobenius Structures and Generalized Deformation of Kodaira Manifolds

Frobenius Structures and Generalized Deformation of Kodaira Manifolds

Holomorphic Koszul-Brylinski homologies of Poisson blow-ups

Holomorphic Koszul–Brylinski homology via Dolbeault cohomology

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