= 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 by
$$
\begin{aligned}
A&=x_1\wedge y_1-x_2\wedge y_2,\\
B&=x_1\wedge x_2+y_1\wedge y_2,\\
C&=x_1\wedge y_2-y_1\wedge x_2.
\end{aligned}
$$
These 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 $*A=-A$, $*B=-B$, and $*C=-C$. By complex linearity,
$$
\boxed{*\alpha=-\alpha.}
$$
This is the fact that <primitive (1,1)-forms on a Kähler surface are anti-self-dual>.
Back to article page
