# Hermitian manifold with harmonic holomorphic volume form

Let M be a compact complex 3-manifold with trivial canonical line bundle and Ω be the non-vanishing holomorphic 3-form.

If the real and imaginary part of Ω are both harmonic with respect to the Hermitian metric g, then is the Hermitian metric kahler ?

No. $\Omega = A + i B$ is closed, so $A$ and $B$ (the real and imaginary parts) are closed. Moreover, with respect to any Hermitian metric $g$, we will have $*_gA = B$ and $*_g B = -A$, so the real and imaginary parts are also co-closed. Thus, the real and imaginary parts of $\Omega$ are harmonic with respect to any Hermitian metric $g$, so this condition gives you no information about the metric $g$.