Let $M$ be a complex manifold and $\Delta^{\bar \partial} = \bar\partial^* \bar\partial + \bar\partial\bar\partial^*$ the complex laplacian. Is it true that $\Delta^{\bar\partial} f = \Delta f$ (the ordinary Hodge laplacian of $f$) for $f\in C^\infty(M)$ a $0$-form?
Is $\Delta^{\bar{\partial}}f = \Delta f$ for $f \in C^{\infty}(M)$?
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riemannian-geometry
complex-geometry