Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an antilinear vector bundle morphism $T_X \to \overline T_X^*$. Then I can fabricate a new $(1,1)$-form on $X$ by setting $F = \alpha \circ \omega^{-1} \circ \overline \beta$.

**Question:** Is the form $F$ harmonic?

I was hoping there was some general theory available to answer this quickly, but I haven't found anything. Calculations in local coordinates also quickly degenerated into filth.

For motivation, if you take $Tr_\omega(F)$ then you get the value of the scalar product on $(1,1)$-forms induced by $\omega$ of $\alpha$ and $\beta$. Thus knowing that $F$ is harmonic lets one conclude that the map $x \mapsto \langle \alpha(x),\beta(x) \rangle_\omega$ is constant.