Holomorphic Poisson Cohomology

Zhuo Chen; Daniele Grandini; Yat-Sun Poon

Complex Manifolds (2015)

  • Volume: 2, Issue: 1, page 34-52, electronic only
  • ISSN: 2300-7443

Abstract

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Holomorphic 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.

How to cite

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Zhuo Chen, Daniele Grandini, and Yat-Sun Poon. "Holomorphic Poisson Cohomology." Complex Manifolds 2.1 (2015): 34-52, electronic only. <http://eudml.org/doc/275919>.

@article{ZhuoChen2015,
abstract = {Holomorphic 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.},
author = {Zhuo Chen, Daniele Grandini, Yat-Sun Poon},
journal = {Complex Manifolds},
keywords = {Poisson cohomology; holomorphic structures; spectral sequence; Dolbeault cohomology; complex manifolds},
language = {eng},
number = {1},
pages = {34-52, electronic only},
title = {Holomorphic Poisson Cohomology},
url = {http://eudml.org/doc/275919},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Zhuo Chen
AU - Daniele Grandini
AU - Yat-Sun Poon
TI - Holomorphic Poisson Cohomology
JO - Complex Manifolds
PY - 2015
VL - 2
IS - 1
SP - 34
EP - 52, electronic only
AB - Holomorphic 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.
LA - eng
KW - Poisson cohomology; holomorphic structures; spectral sequence; Dolbeault cohomology; complex manifolds
UR - http://eudml.org/doc/275919
ER -

References

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