= Nonzero holomorphic top forms are not exact
{title2=$\alpha\ne0\Longrightarrow[\alpha]_{dR}\ne0$}
On a <compact> <complex manifold> of complex dimension $n$, a <holomorphic p-form on a complex manifold> of degree $n$ is $d$-closed. The form $i^{n^2}\alpha\wedge\bar\alpha$ is a nonnegative volume density, positive wherever $\alpha\ne0$: in <holomorphic coordinates> it equals $2^n|f|^2dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n$ for $\alpha=f\,dz_1\wedge\cdots\wedge dz_n$. If $\alpha=d\beta$, its integral would vanish by the <Stokes theorem>, contradicting this positivity. No <Kähler metric> is needed.
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