Primitive (1,1)-forms on a Kähler surface are anti-self-dual
= 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 $*\alpha=-\alpha$, making it an <anti-self-dual differential form>. For a nonzero real primitive harmonic form on a compact surface, $\int\alpha\wedge\alpha=-\|\alpha\|_{L^2}^2<0$.