Non-Kahler manifolds where the different Laplacians are compatible  On a Kahler manifold, the different Laplacians are compatible: $\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
 A: I just add some details to Zhao's answer.
Let $(X,\omega)$ be a hermitian manifold and $\tau$ be the operator of type $(1,0)$ and order $0$ defined by
$$
\tau=[\Lambda_\omega,\partial\omega].
$$
Here $[\bullet,\bullet]$ is the (graded) commutator and $\Lambda_\omega$ the formal adjoint of the operator "wedge product with $\omega$". Often $\partial\omega$ is called the torsion of $\omega$ (which is $\equiv 0$ if and only if $\omega$ is Kähler) and $\tau$ the torsion operator. Then, we have the following identities:
$$
\Delta_{\bar\partial}=\Delta_\partial+[\partial,\tau^\star]-[\bar\partial,\bar\tau^\star],
$$
$$
[\partial,\bar\partial^\star]=-[\partial,\bar\tau^\star],\quad[\bar\partial,\partial^\star]=-[\bar\partial,\tau^\star],
$$
and
$$
\Delta_d=\Delta_\partial+\Delta_{\bar\partial}-[\partial,\bar\tau^\star]-[\bar\partial,\tau^\star].
$$
Therefore, $\Delta_\partial$, $\Delta_{\bar\partial}$ and $\frac 12\Delta_d$ no longer coincide, but they differ by linear differential operator of order $1$ only.
A: Hermitian manifolds $M$ where $$\Delta_d f=2\Delta_{\bar{\partial}} f=2\Delta_{\partial} f$$ holds for every smooth function $f$ on $M$ are called balanced.
For more information, you can search for "balanced hermitian manifolds", Here, for instance, is a paper that reviews their basic properties and conditions to be Kahler.
A: there is a theorem related to this:kahler is equivalent with the laplacian compatible equality above.Thus there no non-Kahler Hermitian manifolds where the above identity holds
