Let $M$ be a complex manifold, and $\omega$ a closed $2$-form. When is $\omega$ a Kahler form? I mean, when does there exist a Kahler metric for which $\omega$ is the corresponding form.

I would (wildly) guess that necessary and sufficient conditions might be got from the K"ahler identities.

Let $M$ be a complex manifold, and $\omega$ a closed $2$-form. When is $\omega$ a Kähler form? I mean, when does there exist a Kähler metric for which $\omega$ is the corresponding form.

I would (wildly) guess that necessary and sufficient conditions might be got from the Kähler identities.