Powers of a compact Kähler form have nonzero cohomology classes ($[{\omega }^k]\ne 0(0\le k\le n)$)

On a nonempty compact Kähler manifold of complex dimension $n$, the Kähler form is closed and ${\int }_X{\omega }^n>0$. If ${\omega }^k=d\eta$ for $1\le k\le n$, then ${\omega }^n=d(\eta \wedge {\omega }^{n-k})$; Stokes theorem contradicts the positive integral. Thus the de Rham cohomology is nonzero in every even degree through $2n$.