Primitive differential form on a Kähler manifold

For degree $k\le n$, a form is primitive if $\Lambda \alpha =0$, equivalently $L^{n-k+1}\alpha =0$. In middle degree $k=n$ this is equivalent to $L\alpha =0$. On a compact Kähler manifold, the same definitions apply to cohomology via harmonic differential forms.

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, $\int \alpha \wedge \alpha =-\Vert \alpha {\Vert }_{L^2}^2<0$.