Poisson cohomology is also called tangent cohomology by Kontsevich [@MR2062626].

As an immediate consequence of Theorem \[extremely\] and Proposition \[Vienna\], we have

If $H$ is an extended Poisson structure on a compact complex manifold ${\boldsymbol{X}}$ and the map ${F}$ (given by Eq. ) is surjective, then all Poisson cohomology groups are finite dimensional.

When $H$ is a holomorphic Poisson bivector field $\pi$, the cochain complex  is the total complex of the double complex as discussed in Corollary 4.26 in [@MR2439547].

On the other hand, if $H=\theta\in{\Omega^{0,1}}({T^{1,0}{\boldsymbol{X}}})$ is a Maurer-Cartan element such that ${\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}-\operatorname{id}$ is invertible, then $\theta$ defines a new complex structure on ${\boldsymbol{X}}$ according to Kodaira [@MR2109686].

The following lemma can be verified directly.

\[Zurich\] Let $H=\theta\in{\Omega^{0,1}}({T^{1,0}{\boldsymbol{X}}})$ be a Maurer-Cartan element such that ${\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}-{\operatorname{id}_{}}$ is invertible. Then the Lie algebroid ${{A}^*}_H$ is isomorphic to $T^{1,0}_\theta{\boldsymbol{X}}\bowtie(T^{0,1}_\theta{\boldsymbol{X}})^*$, where $T^{1,0}_\theta{\boldsymbol{X}}$ and $T^{0,1}_\theta{\boldsymbol{X}}$ are, respectively, the $+{i}$ and $-{i}$ eigenbundles of the deformed almost complex structure $J_\theta:{T{\boldsymbol{X}}}\to{T{\boldsymbol{X}}}$. As a consequence, the differential operator ${{{d}_*}^H}$ in Eq.  is equal to ${\bar{\partial}}_\theta$, the new ${\bar{\partial}}$-operator of the deformed complex structure.

Thus we have

If $H=\theta\in{\Omega^{0,1}}({T^{1,0}{\boldsymbol{X}}})$ is a Maurer-Cartan element such that ${\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}-{\operatorname{id}_{}}$ is invertible, then $${\mathrm{H}^{k}}({\boldsymbol{X}},H)\cong\oplus_{i+j=k}{\mathrm{H}^{i}}({\boldsymbol{X}},\wedge^j T_\theta{\boldsymbol{X}}) ,$$ where $T_\theta{\boldsymbol{X}}$ denotes the holomorphic tangent bundle 