and a Koszul-Brylinski homology for extended Poisson manifolds, and study the relation between them. We extend the notion of coisotropic submanifolds of holomorphic Poisson manifolds to the “extended” setting. We give a criterion on the ellipticity of the complex Lie algebroid (in the sense of Block [@math/0509284]) induced by a [Maurer-Cartan ]{}element. And in the elliptic case, we obtain a duality, which we call Evens-Lu-Weinstein duality, on the Koszul-Brylinski homology groups. As was pointed out in [@arXiv:0903.5065] for the holomorphic Poisson case, this duality generalizes the Serre duality on Dolbeault cohomology.

Note that, modulo gauge equivalences, our extended Poisson structures and Yekutieli’s Poisson deformations (see [@Y1]) are equivalent. It would be interesting to explore the connection between our results on Poisson homology and Berest-Etingof-Ginzburg’s [@MR2034924]. It would also be interesting to investigate if one can extend the method in this paper to study the Bruhat-Poisson structures of Evens-Lu on flag varieties [@EL:flag] and the toric Poisson structures of Caine [@Caine].

**Acknowledgments.** We would like to thank Penn State University (Chen), ETH Zurich (Xu) and Peking University (Stiénon and Xu) for their hospitality while work on this project was being done. We also wish to thank many people for useful discussions 