 Solution

The assertion is pointwise, so choose an oriented orthonormal real coframe $x_1,y_1,x_2,y_2$ compatible with the complex structure, with Kähler form $\omega =x_1\wedge y_1+x_2\wedge y_2$. Real $(1,1)$ forms have a four-dimensional space; imposing $\omega \wedge \alpha =0$ leaves the three-dimensional space spanned byThese are precisely the real primitive differential forms on a Kähler manifold of type $(1,1)$ in dimension two. Complexifying spans all complex primitive $(1,1)$ forms.

Using the complex orientation $x_1\wedge y_1\wedge x_2\wedge y_2$, the Hodge star operator exchanges $x_1\wedge y_1$ with $x_2\wedge y_2$, sends $x_1\wedge x_2$ to $-y_1\wedge y_2$, and exchanges $x_1\wedge y_2$ with $y_1\wedge x_2$. Thus $\ast A=-A$, $\ast B=-B$, and $\ast C=-C$. By complex linearity,This is the fact that primitive (1,1)-forms on a Kähler surface are anti-self-dual.