  In this note, we study the Koszul-Brylinski homology of holomorphic Poisson
manifolds. We show that it is isomorphic to the cohomology of a certain smooth
complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As
a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski
homology is nondegenerate. Finally we compute the Koszul-Brylinski homology for
Poisson structures on $\CP^1\times\CP^1$.
