this pairing is non-degenerate.

Let $({\boldsymbol{X}},H)$ be a compact extended Poisson manifold of complex dimension $n$. Consider the Lie algebroid $B=({{T^{0,1}{\boldsymbol{X}}}}\bowtie{{(T^{1,0}{\boldsymbol{X}})^*}})_H$. Applying Theorem \[Beijing\] and Proposition \[Vienna\], we obtain

\[thm:1\] Let $({\boldsymbol{X}},H)$ be a compact extended Poisson manifold of complex dimension $n$, with $H=\pi+\theta+\omega$. Then the map $${\Omega^{{i,j}}({\boldsymbol{X}})}\otimes{\Omega^{{k,l}}({\boldsymbol{X}})}\to{\mathbb{C}}:
\zeta\otimes\eta\mapsto\int_{{\boldsymbol{X}}}(\zeta\wedge\eta)^{top}$$ induces a pairing on the Koszul-Brylinski Poisson homology: $$\label{eq:pairingPoisson} 
{\mathrm{H}_{k}}({\boldsymbol{X}},H)\otimes{\mathrm{H}_{2n-k}}({\boldsymbol{X}},H)\to{\mathbb{C}}.$$ Moreover, if the bundle map ${F}=(\operatorname{C}+\theta{^{\flat}})\oplus\pi{^{\sharp}}$ maps ${{T^{0,1}{\boldsymbol{X}}}}\oplus{{(T^{1,0}{\boldsymbol{X}})^*}}$ surjectively onto ${{T^{1,0}{\boldsymbol{X}}}}$, then all homology groups ${\mathrm{H}_{\bullet}}({\boldsymbol{X}},H)$ are finite dimensional vector spaces and the pairing  is non-degenerate.

Proof of Theorem \[Thm:main1\]
------------------------------

The following lemmas are needed.

For any $u\in {\Gamma(\wedge^p{A})}$, ${\lambda}\in
{\Omega^{\cdot,\cdot}({\boldsymbol{X}})}$, one has $$\label{Eqt:barpdulambda}
{\bar{\partial}}(u{\cdot}{\lambda})=({\bar{\partial}}u){\cdot}{\lambda}+ (-1)^p u{\cdot}{\bar{\partial}}{\lambda}.$$

For any $u\in{\Gamma(\wedge^p{A})}$, $v\in
{\Gamma(\wedge^q{A})}$, the Schouten bracket ${[u,v]}$ is determined by $$\label{uvSchouten}
{[u,v]}{\cdot}{\lambda}=(-1)^{q+1
}{\lfloor u,{\lfloor v,{\partial}\rfloor}\rfloor}
{\lambda},\quad\forall {\lambda}\in {\Omega^{\bullet,\bullet}({\boldsymbol{X}})}.$$

Both lemmas can be proved by induction; this is left to the reader.

\[lem:pdiHulambdaproperty\] For any $u\in{\Gamma(\wedge^i{A})}$ and ${\lambda}\in{\Omega^{\bullet,\bullet }({\boldsymbol{X}})}$, one has $$\label{Eqt:iHpdulambdaproperty}
{\lfloor {\partial},{\iota_{H}}\rfloor}(u{\cdot}{\lambda})
={[H,u]}{\cdot}{\lambda}+(-1)^{i}u{\cdot}({\lfloor {\partial},{\iota_{H}}\rfloor}{\lambda}) 
.$$ In particular, for any smooth function $f\in{C^{\infty}({\boldsymbol{X}},{\mathbb{C}})}$, one has $$\label{Eqt:iHpdfproperty}
{\lfloor {\partial},{\iota_{H}}\rfloor}(f{\lambda})
={[H,f]}{\cdot}{\lambda}+f{\lfloor {\partial},{\iota_{H}}\rfloor}{\lambda}.$$

According to Eq. , we have $$\begin{aligned}
{[H,u]}{\cdot}{\lambda}=& (-1)^{i+1} {\lfloor H,{\lfloor u,{\partial}\rfloor}\rfloor}{\lambda}\\
=& (-1)^i(u{\cdot}{\partial}(H{\cdot}{\lambda})
-H{\cdot}u{\cdot}({\partial}{\lambda}))
+(H{\cdot}({\partial}(u{\cdot}{\lambda}))
-{\partial}(u{\cdot}H{\cdot}{\lambda})) \\
=& (-1)^i (u{\cdot}{\partial}(H{\cdot}{\lambda})
-u{\cdot}H{\cdot}({\partial}{\lambda}))
+(H{\cdot}({\partial}(u{\cdot}{\lambda}))
-{\partial}(H{\cdot}u{\cdot}{\lambda})) \\
=& -(-1)^{i} u{\cdot}({\lfloor {\partial},{\iota_{H}}\rfloor}{\lambda}) +{\lfloor {\partial},{\iota_{H}}\rfloor}(u{\cdot}{\lambda}) 