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Let $(M,\Omega)$ be a compact Kahler complex manifold of dimension $n$ and let $f,g:M\to \mathbb C$ be two $C^\infty$ functions with the following two propewrties:

  1. $\int_M f\Omega=\int_M g\Omega$

  2. $\Delta_{\bar{\partial}}(f)=\Delta_{\bar{\partial}}(g)$. Where $\Delta_{\bar{\partial}}$ is the usual $\bar{\partial}$-Laplacian constructed by the Hodge $\star$ operator. (However here I suppose that it is enough to assume that $\partial\bar\partial f=\partial\bar\partial g$)

How can I conclude from the above hypotheses that $f=g$?

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Note that $\Delta_{\bar\partial} (f-g) = 0$, this implies that $f-g$ is harmonic and thus constant by compactness. The first condition then implies $f=g$.

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