respect to $({A},{{A}^*})$, i.e. $$\label{Eqt:ExtendedMC}
{\bar{\partial}}H+{\tfrac{1}{2}}{[H,H]}=0 .$$ In this case, $H$ is called an extended Poisson structure.

Any $H\in{\Gamma(\wedge^2{A})}$ decomposes as $$H=\pi+\theta+\omega ,$$ where $\pi\in{\Gamma({T^{2,0}{\boldsymbol{X}}})}$, $\theta\in{\Gamma({{T^{1,0}{\boldsymbol{X}}}}\otimes {{(T^{0,1}{\boldsymbol{X}})^*}})}$ and $\omega\in{\Gamma({(T^{0,2}{\boldsymbol{X}})^*})}$. We will use the following notations to denote the bundle maps induced by natural contraction: $$\begin{aligned}
& \theta{^{\flat}}:~{{T^{0,1}{\boldsymbol{X}}}}\to{{T^{1,0}{\boldsymbol{X}}}},\\
& \theta{^{\sharp}}:~{{(T^{1,0}{\boldsymbol{X}})^*}}\to{{(T^{0,1}{\boldsymbol{X}})^*}},\\
& \pi{^{\sharp}}:~{{(T^{1,0}{\boldsymbol{X}})^*}}\to{{T^{1,0}{\boldsymbol{X}}}},\\
& \omega{^{\flat}}:~{{T^{0,1}{\boldsymbol{X}}}}\to{{(T^{0,1}{\boldsymbol{X}})^*}}.\end{aligned}$$ Note that $\theta{^{\sharp}}=-(\theta{^{\flat}})^*$.

The following lemma is immediate.

An element $H=\pi+\theta+\omega$ is an extended Poisson structure if and only if the following equations are satisfied: $$\begin{aligned}
&{\bar{\partial}}\omega+{[\omega,\theta]}=0, \label{eq:omega}\\
&{\bar{\partial}}\theta+{[\omega,\pi]}+{\tfrac{1}{2}}{[\theta,\theta]}=0, \label{eq:theta}\\
&{\bar{\partial}}\pi + {[\theta,\pi]}=0, \label{eq:pi1}\\
&{[\pi,\pi]}=0. \label{eq:pi2}\end{aligned}$$

\[Lem:MCEqtSpellOut\]

When only one of the three terms of $H$ is not zero, we are left with one of the following three special cases:

1.  $H=\pi$ is an extended Poisson if and only if $\pi$ is a holomorphic Poisson bivector field.

2.  $H=\theta$ is an extended Poisson if and only if ${\bar{\partial}}\theta+{\tfrac{1}{2}}{[\theta,\theta]}=0$. Moreover, if ${\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}-\operatorname{id}$ is invertible, $\theta$ is equivalent to a deformed complex structure [@MR2109686].

3.  $H=\omega$ is an extended Poisson if and only if ${\bar{\partial}}\omega=0$.

In fact, if ${[\omega,\pi]}=0$, Eq.  implies that $\theta$ defines a deformed complex structure (under the assumption that ${\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}-\operatorname{id}$ is invertible). Then, according to Lemma \[Zurich\] below, Eq.  is equivalent to $\bar{\partial}_\theta\omega=0$, where ${\bar{\partial}}_\theta={\bar{\partial}}+[\theta,\cdot]$, and 