Generalized deformations and holomorphic Poisson cohomology of solvmanifolds

AbstractHolomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures.

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A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Complex Manifolds ◽

10.1515/coma-2017-0009 ◽

2017 ◽

Vol 4 (1) ◽

pp. 137-154 ◽

Cited By ~ 1

Author(s):

Yat Sun Poon ◽

John Simanyi

Keyword(s):

Spectral Sequence ◽

Complex Structure ◽

Adjoint Action ◽

Poisson Structures ◽

Dolbeault Cohomology ◽

Poisson Cohomology ◽

Abelian Complex Structure ◽

Nijenhuis Bracket ◽

Holomorphic Poisson Cohomology ◽

Type Decomposition

Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.

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Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

Complex Manifolds ◽

10.1515/coma-2019-0004 ◽

2019 ◽

Vol 6 (1) ◽

pp. 88-102

Author(s):

Yat Sun Poon ◽

John Simanyi

Keyword(s):

Poisson Structure ◽

Algebraic Structure ◽

Complex Structure ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Poisson Cohomology ◽

Abelian Complex Structure ◽

Gerstenhaber Algebras ◽

Holomorphic Poisson Cohomology ◽

Necessary And Sufficient

AbstractIt is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure.We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

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Classical and variational Poisson cohomology

Japanese journal of mathematics ◽

10.1007/s11537-021-2109-2 ◽

2021 ◽

Author(s):

Bojko Bakalov ◽

Alberto De Sole ◽

Reimundo Heluani ◽

Victor G. Kac ◽

Veronica Vignoli

Keyword(s):

Poisson Cohomology

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Poisson cohomology of regular Poisson manifolds

Annales de l’institut Fourier ◽

10.5802/aif.1317 ◽

1992 ◽

Vol 42 (4) ◽

pp. 967-988 ◽

Cited By ~ 24

Author(s):

Ping Xu

Keyword(s):

Poisson Manifolds ◽

Poisson Cohomology

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On Theory of logarithmic Poisson Cohomology

Journal of Mathematics Research ◽

10.5539/jmr.v9n4p209 ◽

2017 ◽

Vol 9 (4) ◽

pp. 209

Author(s):

Joseph Dongho ◽

Alphonse Mbah ◽

Shuntah Roland Yotcha

Keyword(s):

Lie Algebra ◽

Poisson Structure ◽

Commutative Algebra ◽

Linear Representation ◽

Poisson Cohomology ◽

Associative Commutative Algebra

We define the notion of logarithmic Poisson structure along a non zero ideal $\cali$ of an associative, commutative algebra $\cal A$ and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the $\cal A$-module $\Omega_K(\log \cali)$ of logarithmic K\"{a}hler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.

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EQUIVARIANT POISSON COHOMOLOGY AND A SPECTRAL SEQUENCE ASSOCIATED WITH A MOMENT MAP

International Journal of Mathematics ◽

10.1142/s0129167x99000422 ◽

1999 ◽

Vol 10 (08) ◽

pp. 977-1010 ◽

Cited By ~ 13

Author(s):

VIKTOR L. GINZBURG

Keyword(s):

Tensor Product ◽

Spectral Sequence ◽

Vector Fields ◽

Equivariant Cohomology ◽

Poisson Manifold ◽

Momentum Mapping ◽

Poisson Cohomology ◽

Leray Spectral Sequence ◽

Differential Complex ◽

Extensive Introduction

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the E2-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping. An extensive introduction to equivariant cohomology of G-differential complexes is given including some new results and a number of examples and applications are considered.

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