Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

TL;DR

This paper investigates the algebraic properties of holomorphic Poisson cohomology on nilmanifolds with abelian complex structures, establishing conditions for cohomology isomorphisms and algebraic structures.

Contribution

It constructs a canonical non-trivial holomorphic Poisson structure on such nilmanifolds and characterizes when the associated cohomology matches that of the trivial structure.

Findings

Existence of a canonical holomorphic Poisson structure on nilmanifolds with abelian complex structures

Necessary and sufficient conditions for cohomology isomorphism to trivial Poisson cohomology

Conditions for isomorphism at the level of Gerstenhaber algebras

Abstract

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.