We consider holomorphic Poisson structures as a special kind of
generalized geometry in the sense of Hitchin and Gaultieri.
A consideration on local deformation leads us to compute their associated
Lie algebroid cohomology spaces. As this cohomology is represented by the
limit of a bi-complex, we consider various situations early degeneration of
the associated spectral sequence of the bi-complex occurs. Cases for
discussion include Kahlerian manifolds and nilmanifolds with abelian complex
structures.