= Integral pairing of complex top forms
{title2=$i^{n^2}\theta\wedge\overline\theta\geq0$}
For a smooth form $\theta=g\,dz_1\wedge\cdots\wedge dz_n$ of type $(n,0)$, the complex orientation gives $i^{n^2}\theta\wedge\overline\theta=2^n|g|^2\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n$. Consequently its integral on a compact <complex manifold> is positive unless $\theta$ vanishes identically. Applying <Stokes theorem> to $d(\psi\wedge\overline{d\psi})$ proves that a <holomorphic differential form> of degree $n-1$ on a compact complex $n$-manifold is closed, without requiring a <Kähler metric>.
Back to article page
