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.

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

2. If so, given $f$, does the function $\Delta_c(u)=f$ have a solution $u$ if and only if $\int_M\Delta_c(f)=0$?

Many thanks in advance!

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.

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

2. 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!