How to prove that in a Kähler manifold without boundary $\Omega \wedge \cdots \wedge \Omega$ is closed but not exact?

Question

Let $M$ be a compact Kähler manifold without boundary.

1. How can I show that the volume form $$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$ where we have the wedge of $m$ $\Omega$s is closed but not exact for the case that the complex dimension of $M$ is $m$. The hint I am given is to use Stoke's theorem.
2. Also, what happens with the cohomology of this system?

Answer 1

Here is my early thought.

If $M$ is a Kahler manifold, then if $N \subset M$ is a compact complex submanifold without boundary of complex dimension $k$, then the chomology class of $\omega^{k}$ of $H^{2k}(M, \mathbb{R})$ and the homology class of $N$ in $H_{2k}(M, \mathbb{R})$ are non-zero.

Answer 2

Hint Suppose $\Omega^m$ is exact; what does Stokes' Theorem say about the integral $\int_M \Omega^m$?