Holomorphic Poisson Cohomology

TL;DR

This paper explores the cohomology associated with holomorphic Poisson structures, analyzing spectral sequence degeneration conditions on various classes of compact complex manifolds.

Contribution

It identifies necessary conditions for the spectral sequence to degenerate at the second sheet across multiple classes of compact complex manifolds.

Findings

Spectral sequence degenerates on the second sheet for all compact complex surfaces.

Degeneration occurs for Kähler manifolds and certain nilmanifolds.

Provides conditions linking holomorphic Poisson cohomology to manifold geometry.

Abstract

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is 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, K\"ahler manifolds, and nilmanifolds with abelian complex structures or complex parallelizable manifolds.