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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access August 4, 2017

A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

Yat Sun Poon EMAIL logo and John Simanyi
From the journal Complex Manifolds

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.

MSC 2010: 53D18; 53D17; 32G20; 18G40; 14D07

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Received: 2017-6-16
Accepted: 2017-7-12
Published Online: 2017-8-4
Published in Print: 2017-2-23

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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