\[Thm:main1\].

We adopt an inductive approach. First we prove the commutativity of Diagram  for $k=0$.

Note that for any $f\in{C^{\infty}({\boldsymbol{X}},{\mathbb{C}})}$, $u\in{\Gamma(\wedge^k{A})}$ and $s\in{\Gamma({\mathscr{L}})}$, one has $$\begin{aligned}
{\tau}{{\breve{{d}}}_*^H}(f u\otimes s) &= {\tau}\big(f{{\breve{{d}}}_*^H}(u\otimes s)
+(({\bar{\partial}}f+{[H,f]})\wedge u)\otimes s\big) & \text{by Eq.~\eqref{Eqt:bdeesHuotimess}} \\
&= f{\tau}{{\breve{{d}}}_*^H}(u\otimes s)+({\bar{\partial}}f+{[H,f]}){\cdot}{\tau}(u\otimes s) . &\end{aligned}$$

On the other hand, if we write ${\lambda}={\tau}(u\otimes s)$, one has $$\begin{aligned}
& ({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor}){\tau}(fu\otimes s) & \\
=& ({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})(f{\lambda}) & \\
=& {\bar{\partial}}f\wedge{\lambda}+f{\bar{\partial}}{\lambda}+{[H,f]}{\cdot}{\lambda}+f{\lfloor {\partial},{\iota_{H}}\rfloor}{\lambda}& \text{by Eq.~\eqref{Eqt:iHpdfproperty}} \\
=&f({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor}){\tau}(u\otimes s)
+({\bar{\partial}}f+{[H,f]}){\cdot}{\tau}(u\otimes s). &\end{aligned}$$

It thus follows that the map ${\tau}\circ{{\breve{{d}}}_*^H}-({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})\circ{\tau}$ is ${C^{\infty}({\boldsymbol{X}})}$-linear. Take a local holomorphic chart $(z^1,\dots,z^n)$ and write $H$ locally as in Eq.  in Lemma \[Lem:localstrAbdsH\]. Again take $s$ as in Eq. . For $k=0$, we have ${{\breve{{d}}}_*^H}s={{\mathcal{D}}^H}s$, which is given locally by Eq. . Then, we compute $$\begin{aligned}
{\tau}({{\breve{{d}}}_*^H}s) =& (2\tfrac{\partial{\pi^{i,p}}}{\partial z^p}{{\tfrac{\partial}{\partial{z^{i}}}}} -\tfrac{\partial{\theta^{p}_{i}}}{\partial z^p}{d \bar{z}^{i}}){\cdot}({d z^{1}}\wedge\cdots\wedge{d z^{n}}) \\
=& 2\sum_{i=1}^n(-1)^{i+1}\tfrac{\partial{\pi^{i,p}}}{\partial z^p} {d z^{1}}\wedge\cdots\wedge\widehat{{d z^{i}}}\wedge\cdots\wedge{d z^{n}}
-\tfrac{\partial{\theta^{p}_{i}}}{\partial z^p}{d \bar{z}^{i}}\wedge{d z^{1}}\wedge\cdots\wedge{d z^{n}}
.\end{aligned}$$ Thus we have $$\begin{aligned}
({\bar{\partial}}+{\lfloor {\partial},{\iota_{H}}\rfloor})s =& {\partial}{\iota_{H}}({d z^{1}}\wedge\cdots\wedge{d z^{n}}) \\
=& {\partial}\Big(2\sum_{i<j}(-1)^{i+j-1}{\pi^{i,j}}{d z^{1}}\wedge\cdots\wedge
\widehat{{d z^{i}}}\wedge\cdots\wedge\widehat{{d z^{j}}}\wedge\cdots\wedge{d z^{n}} \\
& \quad +\sum_{p=1}^n(-1)^{p+1}{\theta^{p}_{i}}{d \bar{z}^{i}}
\wedge{d z^{1}}\wedge\cdots\wedge\widehat{{d z^{p}}}\wedge\cdots\wedge{d z^{n}} \\
& \quad -{\omega_{k,l}}{d \bar{z}^{k}}\wedge{d \bar{z}^{l}}\wedge
{d z^{1}}\wedge\cdots\cdots\wedge{d z^{n}}\Big) \\
=& {\tau}({{\breve{{d}}}_*^H}s)
.\end{aligned}$$ It thus follows that Diagram  indeed commutes when 