  Through the theory of Lie bi-algebroids and generalized complex structures,
one could define a cohomology theory naturally associated to a holomorphic
Poisson structure. It is known that it is the hypercohomology of a bi-complex
such that one of the two operators is the classical
$\overline{\partial}$-operator. Another operator is the adjoint action of the
Poisson bivector with respect to the Schouten-Nijenhuis bracket. The
hypercohomology is naturally computed by one of the two associated spectral
sequences. In a prior publication, the author of this article and his
collaborators investigated the degeneracy of this spectral sequence on the
second page. In this note, the author investigates the conditions for which
this spectral sequence degenerates on the first page. Particular effort is
devoted to nilmanifolds with abelian complex structures.
