A Hodge-Type Decomposition of Holomorphic Poisson Cohomology on Nilmanifolds

TL;DR

This paper explores conditions under which the spectral sequence for holomorphic Poisson cohomology degenerates on nilmanifolds with abelian complex structures, leading to a Hodge-type decomposition for certain Poisson structures.

Contribution

It establishes a criterion for spectral sequence degeneration on nilmanifolds and derives a Hodge-type decomposition for holomorphic Poisson cohomology in this setting.

Findings

Spectral sequence degenerates on the first page for certain nilmanifolds.

Hodge-type decomposition of holomorphic Poisson cohomology obtained.

Examples provided for 2-step nilmanifolds.

Abstract

A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-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.