the twisted Dolbeault cohomology.

3.  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 Lemma \[Zurich\], the cochain complex $(\bigoplus_{i-j=n-k}{\Omega^{i,j}({\boldsymbol{X}})},{\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})$ is isomorphic to $(\bigoplus_{i-j=n-k}\Omega_\theta^{i,j}({\boldsymbol{X}}),\;\bar{\partial}_\theta)$, where $\bar{\partial}_\theta$ is the $\bar{\partial}$-Dolbeault operator of the deformed complex structure. As a consequence, we have ${\mathrm{H}_{k}}({\boldsymbol{X}},\theta){\cong}\oplus_{j-i=n-k}{\mathrm{H}^{i,j}}_\theta({\boldsymbol{X}})$, where ${\mathrm{H}^{i,j}}_\theta({\boldsymbol{X}})$ is the Dolbeault cohomology of the deformed complex structure.

Evens-Lu-Weinstein duality
--------------------------

Consider a compact complex (and therefore orientable) manifold ${\boldsymbol{X}}$ with $\dim_{{\mathbb{C}}}{\boldsymbol{X}}=n$, a complex Lie algebroid $B$ over ${\boldsymbol{X}}$ with $\operatorname{rk}_{{\mathbb{C}}}B=r$. According to [@MR1726784], the complex line bundle ${Q_{{B}}}=\wedge^r B\otimes\wedge^{2n}{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$ is a module over the complex Lie algebroid $B$. If ${{Q_{{B}}}^{{\frac{1}{2}}}}$ exists as a complex vector bundle, ${{Q_{{B}}}^{{\frac{1}{2}}}}$ becomes a $B$-module as well. There is a natural map $$\phi: {\Gamma(\wedge^k B^*\otimes{{Q_{{B}}}^{{\frac{1}{2}}}})}
\otimes {\Gamma(\wedge^{r-k} B^*\otimes{{Q_{{B}}}^{{\frac{1}{2}}}})} 
\to {\Gamma(\wedge^r B^*\otimes{Q_{{B}}})}
{\cong}{\Gamma(\wedge^{2n}T^*_{\mathbb{C}}{\boldsymbol{X}})}$$

Integrating, we get the pairing $$\label{eq:pairing} 
{\Gamma(\wedge^k B^*\otimes{{Q_{{B}}}^{{\frac{1}{2}}}})} \otimes
{\Gamma(\wedge^{r-k} B^*\otimes{{Q_{{B}}}^{{\frac{1}{2}}}})}\to{\mathbb{C}},  \qquad
\xi\otimes\eta\mapsto\int_{{\boldsymbol{X}}}\phi(\xi\otimes\eta) 
.$$

The following result is essentially due to Evens-Lu-Weinstein [@MR1726784] for the pairing, and to Block [@math/0509284] for the non-degeneracy (see also [@arXiv:0903.5065]).

\[Beijing\] For a complex Lie algebroid $B$, with $\operatorname{rk}_{{\mathbb{C}}}B=r$, over a compact manifold ${\boldsymbol{X}}$, the pairing  induces a pairing $${\mathrm{H}^{k}}(B,{{Q_{{B}}}^{{\frac{1}{2}}}})\otimes{\mathrm{H}^{r-k}}(B,{{Q_{{B}}}^{{\frac{1}{2}}}})\to{\mathbb{C}}.$$ Moreover, if $B$ is an elliptic Lie algebroid, 