$k=0$.

Now assume that we have proved the commutativity of Diagram  when $k\leq m$ (where $0\leq m\leq 2n-1$). To prove the $k=m+1$ case, we consider a section $(u\wedge w)\otimes s\in{\Gamma(\wedge^{m+1}{A}\otimes{\mathscr{L}})}$, where $u\in{\Gamma({A})}$, $w\in{\Gamma(\wedge^m{A})}$ and $s\in{\Gamma({\mathscr{L}})}$. Then $$\begin{aligned}
& ({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor}){\tau}((u\wedge w)\otimes s) & \\
=& ({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})(u{\cdot}{\lambda})
& \text{where }{\lambda}=w{\cdot}s \\
=& {\bar{\partial}}u{\cdot}{\lambda}-u{\cdot}{\bar{\partial}}{\lambda}+{[H,u]}{\cdot}{\lambda}-u{\cdot}({\lfloor {\partial},{\iota_{H}}\rfloor}{\lambda})
& \text{by Eqs.~\eqref{Eqt:barpdulambda} and~\eqref{Eqt:iHpdulambdaproperty}} \\
=& {\bar{\partial}^H}u{\cdot}{\lambda}-u{\cdot}({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor}){\lambda}\\
=& {\tau}\big(({\bar{\partial}^H}u\wedge w)\otimes s\big)-u{\cdot}{\tau}{{\breve{{d}}}_*^H}(w\otimes s)
& \text{by assumption} \\
=& {\tau}{{\breve{{d}}}_*^H}((u\wedge w)\otimes s) . &\end{aligned}$$ This concludes the proof.

Modular classes
---------------

The modular class of a Lie algebroid was introduced by Evens-Lu-Weinstein [@MR1726784]. The following version for complex Lie algebroids appeared in the arXiv version `dgga/9610008` of [@MR1726784] but not in the published paper. It is also implied in [@math/0703298]. The presentation which we give below was communicated to us by Camille Laurent-Gengoux [@LaurentLetter].

Let $B$ be a complex Lie algebroid over a real manifold $M$, with $\operatorname{rk}_{{\mathbb{C}}}B=r$ and $\dim M=m$. Its Evens-Lu-Weinstein module is ${Q_{{B}}}=\wedge^{r}B\otimes\wedge^{m}T^*_{\mathbb{C}}M$.

Consider the complex of sheaves $$\label{Rome} {\mathcal{\widetilde{S}}^{0} }
\stackrel{\tilde{d}_B}{\longrightarrow} {\mathcal{S}^{1} }
\stackrel{{d}_B}{\longrightarrow} {\mathcal{S}^{2} } \cdots
\stackrel{{d}_B}{\longrightarrow} {\mathcal{S}^{r} },$$ where ${\mathcal{\widetilde{S}}^{0} }$ is the sheaf of nowhere vanishing smooth complex valued functions on $M$; ${\mathcal{S}^{\bullet} }$ is the sheaf of sections of 