Suppose $(M, J)$ is a compact complex manifold of complex dimension $n$ and there exists a Kaehler metric $\Omega$ on $M$. By the global $\partial \bar\partial$ lemma, any Kaehler metric in the same cohomology class as $\Omega$ is of the form $\Omega' := \Omega + \partial\bar\partial f$ for $f \in \mathcal{C}^\infty(M)$. I am trying to understand why the set of Kaehler metrics in the same cohomology class as $\Omega$ is in 1-to-1 correspondence with the set of functions $f \in \mathcal{C}^\infty(M)$ such that
- $(\Omega + \partial \bar\partial f)(\cdot, J \cdot)$ is a positive definite symmetric tensor and
- $\int_M f \Omega^n = 0$
I get that the first condition is necessary, since otherwise the form $\Omega + \partial\bar\partial f$ wouldn't come from a Hermitian metric, hence it wouldn't be Kaehler.
But why is the second condition either necessary or sufficient? For example, I don't know how I would intuitively arrive at the fact that (2) is necessary given that $\Omega'$ is Kaehler.
If you add a constant to $f$, it doesn't change the Kähler form. So restricting to functions with zero average value is a way to ensure that the map from functions to Kähler metrics is injective.