  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 $E_2$.
