element is an $$H=\pi+\theta+\omega
\in{\Omega^{0,0}}(\wedge^2 {T^{1,0}{\boldsymbol{X}}})\oplus{\Omega^{0,1}}(\wedge^1
{T^{1,0}{\boldsymbol{X}}})\oplus{\Omega^{0,2}}(\wedge^0 {T^{1,0}{\boldsymbol{X}}})$$ (where ${\Omega^{0,p}}(\wedge^q
{T^{1,0}{\boldsymbol{X}}}$) denotes the space of $\wedge^q{T^{1,0}{\boldsymbol{X}}}$-valued $(0,p)$-forms on ${\boldsymbol{X}}$) satisfying the following equations: $$\begin{aligned}
&{\bar{\partial}}\omega+{[\omega,\theta]}=0,  &{\bar{\partial}}\pi + {[\theta,\pi]}=0,  \\
&{\bar{\partial}}\theta+{[\omega,\pi]}+{\tfrac{1}{2}}{[\theta,\theta]}=0,  &{[\pi,\pi]}=0.\end{aligned}$$

Holomorphic Poisson bivector fields are special cases of such [Maurer-Cartan ]{}elements, as are holomorphic $(0,2)$-forms. For this reason, complex manifolds endowed with such a [Maurer-Cartan ]{}element $H$ will be called extended Poisson manifolds. In a recent paper [@arXiv:0903.5065], one of the authors studied the Koszul-Brylinski homology of holomorphic Poisson manifolds, and established a duality on it using the general theory developed by Evens-Lu-Weinstein [@MR1726784].

In this paper, in order to study the geometry of extended Poisson manifolds, we apply the Evens-Lu-Weinstein theory to complex Lie algebroids. Indeed, considering Maurer-Cartan elements as Hamiltonian operators (in the sense of [@MR1472888]) deforming a Lie bialgebroid [@MR1262213], we define a complex Lie algebroid, which mimics the role played by the cotangent Lie algebroid in real Poisson geometry. It is not surprising that, for a holomorphic Poisson structure, this complex Lie algebroid is the derived Lie algebroid of the holomorphic cotangent Lie algebroid $({T^*{\boldsymbol{X}}})_{\pi}$, i.e. the matched pair ${{T^{0,1}{\boldsymbol{X}}}}\bowtie ({T^*{\boldsymbol{X}}})_{\pi}^{(1,0)}$ studied in [@MR2439547; @arXiv:0903.5065]. Using this complex Lie algebroid, we introduce a Lichnerowicz-Poisson cohomology 