# A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Complex Manifolds (2017)

- Volume: 4, Issue: 1, page 137-154
- ISSN: 2300-7443

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topYat Sun Poon, and John Simanyi. "A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds." Complex Manifolds 4.1 (2017): 137-154. <http://eudml.org/doc/288472>.

@article{YatSunPoon2017,

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.},

author = {Yat Sun Poon, John Simanyi},

journal = {Complex Manifolds},

keywords = {Holomorphic Poisson; Cohomology; Hodge theory; Nilmanifolds},

language = {eng},

number = {1},

pages = {137-154},

title = {A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds},

url = {http://eudml.org/doc/288472},

volume = {4},

year = {2017},

}

TY - JOUR

AU - Yat Sun Poon

AU - John Simanyi

TI - A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

JO - Complex Manifolds

PY - 2017

VL - 4

IS - 1

SP - 137

EP - 154

AB - 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.

LA - eng

KW - Holomorphic Poisson; Cohomology; Hodge theory; Nilmanifolds

UR - http://eudml.org/doc/288472

ER -

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