$\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$ for a global holomorphic p-form $\alpha$

Question

Let $X$ be a compact kahler manifold. And $\alpha\in H^{p,0}(X)$. How to see $\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$?

Answer 1

Note that a $(p,0)$-class is primitive, since the adjoint Lefschetz operator $\Lambda$ lowers bidegree by $(-1,-1)$. Since $X$ is compact Kähler, the Hodge-Riemann bilinear relations imply that $\int \alpha \wedge \bar\alpha \wedge \omega^{n-p} \neq 0$, so clearly the cohomology class of $\alpha \wedge \bar\alpha$ cannot vanish.