A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

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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Holomorphic Poisson Cohomology

Complex Manifolds ◽

10.1515/coma-2015-0005 ◽

2015 ◽

Vol 2 (1) ◽

Cited By ~ 3

Author(s):

Zhuo Chen ◽

Daniele Grandini ◽

Yat-Sun Poon

Keyword(s):

Spectral Sequence ◽

Complex Structure ◽

Complex Structures ◽

Dolbeault Cohomology ◽

Double Complex ◽

Poisson Cohomology ◽

Compact Complex Surfaces ◽

Generalized Geometry ◽

Holomorphic Poisson Cohomology ◽

Compact Complex Manifolds

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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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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TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x9500200x ◽

1995 ◽

Vol 10 (30) ◽

pp. 4325-4357 ◽

Cited By ~ 24

Author(s):

A. JOHANSEN

Keyword(s):

Complex Structure ◽

Kähler Manifold ◽

Dolbeault Cohomology ◽

Kähler Metric ◽

Yang Mills ◽

Kahler Manifold ◽

Brst Charge ◽

Kahler Metric ◽

Compact Kähler Manifold ◽

Topological Theories

It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

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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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Dolbeault cohomology of nilmanifolds with left-invariant complex structure

Complex and Differential Geometry - Springer Proceedings in Mathematics ◽

10.1007/978-3-642-20300-8_18 ◽

2011 ◽

pp. 369-392 ◽

Cited By ~ 7

Author(s):

Sönke Rollenske

Keyword(s):

Complex Structure ◽

Dolbeault Cohomology ◽

Invariant Complex Structure

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Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence

Complex Manifolds ◽

10.1515/coma-2020-0008 ◽

2020 ◽

Vol 7 (1) ◽

pp. 141-144

Author(s):

Aleksandar Milivojević

Keyword(s):

Spectral Sequence ◽

Lie Group ◽

Duality Theorem ◽

Complex Structure ◽

Short Note ◽

Almost Complex Structure ◽

Analogous Statement ◽

Almost Complex ◽

Hodge Numbers ◽

Frölicher Spectral Sequence

AbstractSerre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies \dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” \dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].

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GRADED POISSON STRUCTURES AND SCHOUTEN–NIJENHUIS BRACKET ON ALMOST COMMUTATIVE ALGEBRAS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500429 ◽

2012 ◽

Vol 09 (05) ◽

pp. 1250042 ◽

Cited By ~ 1

Author(s):

FERDINAND NGAKEU

Keyword(s):

Poisson Bracket ◽

Abelian Groups ◽

Poisson Manifolds ◽

Poisson Structures ◽

Symplectic Structures ◽

Commutative Algebras ◽

Nijenhuis Bracket

We introduce and study the notion of abelian groups graded Schouten–Nijenhuis bracket on almost commutative algebras and show that any Poisson bracket on such algebras is defined by a graded bivector as in the classical Poisson manifolds. As a particular example, we introduce and study symplectic structures on almost commutative algebras. Our result is a generalization of the ℤ2-graded (super)-Poisson structures.

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