Powers of a compact Kähler form have nonzero cohomology classes
= Powers of a compact Kähler form have nonzero cohomology classes
{title2=$[\omega^k]\ne0\quad(0\leq k\leq n)$}
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 $1\leq k\leq n$, 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 $2n$.