of the deformed complex manifold ${\boldsymbol{X}}$.

Coisotropic submanifolds
------------------------

Suppose that ${\boldsymbol{Y}}\subseteq {\boldsymbol{X}}$ is a complex submanifold [@MR2109686]. Set $$\begin{gathered}
{N^{1,0}{\boldsymbol{Y}}}={\left\{\xi\in({{T^{1,0}{\boldsymbol{X}}}}|_{{\boldsymbol{Y}}})^* {\;\text{s.t.}\;}{\langle \xi | Y\rangle}=0,\;\forall Y\in{T^{1,0}{\boldsymbol{Y}}}\right\}} ,\end{gathered}$$ and consider the subbundle ${K}={T^{0,1}{\boldsymbol{Y}}}\oplus{N^{1,0}{\boldsymbol{Y}}}$ of ${{A}^*}$.

A complex submanifold ${\boldsymbol{Y}}$ of ${\boldsymbol{X}}$ is called **coisotropic** if $H(u,v)=0$, for all $u,v\in{K}$.

If $H=\pi$ is a holomorphic Poisson bivector field, then ${\boldsymbol{Y}}$ is coisotropic if and only if it is coisotropic in the usual sense, i.e. ${\pi}{(\xi_1,\xi_2)}=0$, $\forall\xi_1,\xi_2\in{N^{1,0}{\boldsymbol{Y}}}$, or $\pi{^{\sharp}}({N^{1,0}{\boldsymbol{Y}}})\subseteq{T^{1,0}{\boldsymbol{Y}}}$.

If $H=\omega$, then ${\boldsymbol{Y}}$ is coisotropic if and only if $\iota^*\omega=0$, where $\iota:{\boldsymbol{Y}}\to{\boldsymbol{X}}$ is the embedding map.

If $H=\theta$, then ${\boldsymbol{Y}}$ is coisotropic if and only if $\theta{^{\flat}}({T^{0,1}{\boldsymbol{Y}}})\subseteq{T^{1,0}{\boldsymbol{Y}}}$.

It is well known that given a coisotropic submanifold $C$ of a real Poisson manifold $(P,\pi)$, the conormal bundle $NC={\left\{\xi\in T_c^*P {\;\text{s.t.}\;}c\in C;\; {\langle \xi | X\rangle}=0,\;\forall X\in T_c C\right\}}$ is a Lie subalgebroid of the cotangent Lie algebroid $(T^*P)_\pi$ [@MR959095]. The following proposition can be considered as an analogue of this fact in the extended Poisson setting.

Let ${\boldsymbol{Y}}$ be a coisotropic submanifold of the extended Poisson manifold $({\boldsymbol{X}},H)$. Then the vector subbundle ${K}={T^{0,1}{\boldsymbol{Y}}}\oplus{N^{1,0}{\boldsymbol{Y}}}$ is a Lie subalgebroid of ${{A}^*}_H$. That is, $a_*^H$ maps ${K}$ into $T_{{\mathbb{C}}}{\boldsymbol{Y}}$ and for any smooth extensions $\widetilde{u},\widetilde{v}\in{\Gamma({{A}^*}_H)}$ to ${\boldsymbol{X}}$ 