  The purpose of this paper is to study the properties of holomorphic Poisson
manifolds $(M,\pi)$ under the assumption of $\partial_{}\bar{\partial}$--lemma
or $\partial_{\pi}\bar{\partial}$--lemma. Under these assumptions,we show that
the Koszul--Brylinski homology can be recovered by the Dolbeault cohomology,
and prove that the DGLA
$(A_M^{\bullet,\bullet},\bar{\partial},[-,-]_{\partial_\pi})$ is
formal.Furthermore,we discuss the Maurer--Cartan elements of
$(A_M^{\bullet,\bullet}[[t]],\bar{\partial},[-,-]_{\partial_\pi})$ which induce
the deformations of complex structure of $M$.
