TITLE: When is a Form a Kähler Form?
QUESTION [15 upvotes]: 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.

REPLY [8 votes]: It's possible to approach the question in a slightly different way if $X$ is compact. Donu and Spiro are of course right in that the condition for a smooth closed $(1,1)$-form $\omega$ to be a Kahler metric is that $\omega$ be positive. This is a pointwise condition on the manifold $X$, which can be difficult to check unless we have quite explicit expressions for $\omega$.
Suppose for a minute that $\omega$ is a Kahler metric. If $Y \subset X$ is a $p$-dimensional closed subspace, then we see that
$$
\int_Y \omega^p = \int_Y [\omega]^p > 0,
$$
where $[\omega]$ is the cohomology class that $\omega$ defines. This holds because $\omega$ induces a Kahler metric on (the smooth part of) $Y$ and $\omega^p$ is $p!$ times the volume form of the induced metric. (I'm skipping over some subtle things that involve integration over singular subspaces, but one can make sense of all of this.)
Demailly and Paun proved that the converse holds. That is, if we have a cohomology class $[\omega]$ such that its integral over any subspace of $X$ (including $X$ itself) is positive, then $[\omega]$ contains a Kahler metric. (This is a big generalization of the Nakai-Moishezon criterion in algebraic geometry.)
This doesn't help if what you're interested in is knowing that the precise form you have is Kahler, because given a Kahler form we can always take a nonconstant real smooth function $f$ and add $i\partial\bar\partial f$ to the form; this doesn't change the cohomology class and adding (or subtracting) giant such multiples will break the positivity of the resulting form. But for some questions, all that matters is the cohomology class of the Kahler form, and for those you can check positivity of the class by integration.