  We study holomorphic Poisson manifolds and holomorphic Lie algebroids from
the viewpoint of real Poisson geometry. We give a characterization of
holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of
Magri-Morosi and describe a double complex which computes the holomorphic
Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle
$A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids
$(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid
cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie algebroid
$T^{0,1}X\bowtie A^{1,0}$. In the case when $(X,\pi)$ is a holomorphic Poisson
manifold and $A=(T^*X)_\pi$, such an elliptic Lie algebroid coincides with the
Dirac structure corresponding to the associated generalized complex structure
of the holomorphic Poisson manifold.
