Holomorphic Poisson Structures and its Cohomology on Nilmanifolds

TL;DR

This paper studies holomorphic Poisson structures on nilmanifolds with abelian complex structures, showing cohomology is isomorphic to invariant forms, and analyzes the degeneration of the associated spectral sequence.

Contribution

It establishes the isomorphism of Dolbeault cohomology with invariant form cohomology and investigates the degeneration behavior of the Poisson bi-complex spectral sequence.

Findings

Cohomology with coefficients in holomorphic polyvector fields is isomorphic to invariant form cohomology.

Spectral sequence of the Poisson bi-complex degenerates at E_2 for certain structures.

Examples where the spectral sequence does not degenerate at E_2 are provided.

Abstract

The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in holomorphic polyvector fields is isomorphic to the cohomology of invariant forms with coefficients in invariant polyvector fields. We then quickly identify the existence of invariant holomorphic Poisson structures. More important, the spectral sequence of the Poisson bi-complex associated to such holomorphic Poisson structure degenerates at $E_2$. We will also provide examples of holomorphic Poisson structures on such manifolds so that the related spectral sequence does not degenerate at $E_2$.