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?



Non-Kahler manifolds where the different Laplacians are compatible 
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](http://cdsweb.cern.ch/record/422889/files/), for instance, is a paper that reviews their basic properties and conditions to be Kahler.


