A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Yat 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 -