  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.
