  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.
