Primitive (1,1)-forms on a Kähler surface are anti-self-dual

On a complex two-dimensional Kähler manifold, a primitive $(1,1)$ form satisfies $\ast \alpha =-\alpha$, making it an anti-self-dual differential form. For a nonzero real primitive harmonic form on a compact surface, $\int \alpha \wedge \alpha =-\Vert \alpha {\Vert }_{L^2}^2<0$.