A question about complex Laplacian on compact Hermitian manifolds Let $(M,g)$ be a complex $n$-dim compact connected Hermitian manifold, and $\Delta_c(\cdot):=g^{i\bar{j}}\partial_i\partial_{\bar{j}}(\cdot)$  the complex Laplacian acting on smooth functions on $M$. It is well-known that this $\Delta_c$ is in general not equal to the usual Laplacian and this holds exactly when the metric $g$ is balanced. (due to Gauduchon?)
My question is, does this complex Laplacian $\Delta_c$ behave like the usual Laplacian? To be more precise, I have two related questions.

*

*Does $\int_M\Delta_c(f)=0?$ for any $f$?


*If so, given $f$, does the function $\Delta_c(u)=f$ have a solution $u$ if and only if $\int_M f=0$?
Many thanks in advance!
 A: For your first question, note that (let $\omega$ be the Hermitian form)
$$\int_M\Delta_c(f) \omega^n=\int_M(\mathrm{tr}_\omega \sqrt{-1}\partial
\overline{\partial}f)\omega^n=n\int_M \sqrt{-1}\partial
\overline{\partial}f \wedge \omega^{n-1}$$
Using Stoke's formula, we deduce that
$$\int_M \partial
\overline{\partial}f \wedge \omega^{n-1}=\int_M \overline{\partial}f \wedge \mathrm{d}\omega^{n-1}=\int_M \overline{\partial}f \wedge \partial \omega^{n-1}=\int_M f \wedge \partial \overline{\partial}  \omega^{n-1}$$
Then
$$
\int_M\Delta_c(f) \omega^n=n \int_M f \wedge \sqrt{-1}\partial \overline{\partial}  \omega^{n-1}
$$
And $\int_M\Delta_c(f) \omega^n=0$ for all $f$ iff $\partial \overline{\partial}  \omega^{n-1}=0$. Such metric is called a $\textbf{Gauduchon metric}$.
The answer to your second question is positive and it was first proved by Gauduchon in "Le théorème de l'excentricité nulle. C. R. Acad. Sci. Paris 285, 387–390 (1977)".
A more accessible reference is "The Monge–Ampère equation for non-integrable almost complex structures" by Chu, Tosatti and Weinkove. They proved the results in Theorem 2.1, Theorem 2.2 in their paper (for almost Hermitian manifolds):

