in the r.h.s. $X\in\wedge^k{A}$ is regarded as an element of the Clifford algebra ${\mathcal{C}({\boldsymbol{E}})}$ and $s\in{\mathscr{L}}$ is regarded as an element in $\wedge^\bullet{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$, defines an isomorphism of vector bundles $${\tau}:\wedge^k{A}\otimes{\mathscr{L}}\to\bigoplus_{i-j=n-k}{(T^{i,j}{\boldsymbol{X}})^*} .$$

Equivalently, $${\tau}\big((W\wedge\xi)\otimes s\big)=(-1)^{\frac{w(w-1)}{2}}{\iota_{W}}(\xi\wedge s)
=(-1)^{\frac{w(w-1)}{2}+n(k-w)}({\iota_{W}}s)\wedge\xi ,$$ for $W\in{T^{w,0}{\boldsymbol{X}}}$, $\xi\in{(T^{0, k-w}{\boldsymbol{X}})^*}$ and $s\in {\mathscr{L}}$.

We define the inner product of $H\in{\Gamma(\wedge^2{A})}$ with ${\lambda}\in{\Gamma(\wedge^\bullet{T^*_{{\mathbb{C}}}{\boldsymbol{X}}})}$ as $${\iota_{H}}{\lambda}= -H {\cdot}{\lambda}.$$ This coincides with the usual inner product of bivector fields with differential forms. Introduce $${\lfloor {\partial},{\iota_{H}}\rfloor}= {{\partial}}\circ
{{\iota_{H}}}-{{\iota_{H}}}\circ {{\partial}}
:~{\Gamma(\wedge^\bullet {T^*_{{\mathbb{C}}}{\boldsymbol{X}}})}\to
{\Gamma(\wedge^\bullet {T^*_{{\mathbb{C}}}{\boldsymbol{X}}})} .$$

Let us denote ${\Omega^{i,j}({\boldsymbol{X}})}={\Gamma({(T^{i,j}{\boldsymbol{X}})^*})}$. The following theorem is the main result in this section.

\[Thm:main1\] The diagram $$\label{Diagram:stdIso}
\xymatrix{ {\Gamma(\wedge^k{A}\otimes{\mathscr{L}})} \ar[rr]^{{{\breve{{d}}}_*^H}} \ar[d]_{{\tau}} & & 
{\Gamma(\wedge^{k+1}{A}\otimes{\mathscr{L}})} \ar[d]^{{\tau}} \\ 
\bigoplus_{i-j=n-k} {\Omega^{i,j}({\boldsymbol{X}})} \ar[rr]_{{\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor}} & & \bigoplus_{i-j=n-k-1} {\Omega^{i,j}({\boldsymbol{X}})} }$$ commutes.

The cohomology of the cochain complex $(\bigoplus_{i-j=n-k}{\Omega^{i,j}({\boldsymbol{X}})},{\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})$ is called the **Koszul-Brylinski Poisson homology** of the extended Poisson manifold $({\boldsymbol{X}},H)$, and denoted ${\mathrm{H}_{\bullet}}({\boldsymbol{X}},H)$.

1.  If $H=\pi$ is a holomorphic Poisson bivector field, the cochain complex $(\bigoplus_{i-j=n-k}{\Omega^{i,j}({\boldsymbol{X}})},{\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})$ is the total complex of a double complex. Its cohomology is the usual Koszul-Brylinski Poisson homology of a holomorphic Poisson manifold, as studied in detail by one of the authors [@arXiv:0903.5065].

2.  If $H=\omega\in{\Omega^{0,2}({\boldsymbol{X}})}$ with $\bar{\partial}\omega=0$, the complex $(\bigoplus_{i-j=n-k}{\Omega^{i,j}({\boldsymbol{X}})},{\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})$ becomes $(\bigoplus_{i-j=n-k}{\Omega^{i,j}({\boldsymbol{X}})},{\bar{\partial}}+({\partial}\omega)\wedge)$. Its cohomology is 