Pome3AA98220C897Lemma 5.38.1.3AA98220C897 Let $M$ be a complex manifold, let $\beta $ be a smooth real $(1,1)$-form on $M$, let $w\in \mathrm{PSH}(M,\beta )$, and let $E_1,\ldots ,E_ N$ be distinct prime divisors on $M$, with $N\geq 1$. Then
The form $\beta $ need not be closed.
Proof. Work on a coordinate ball and choose a smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \beta $. The current
is closed and positive. A smooth form, viewed through its coefficient measures, assigns zero mass to every proper analytic set. Thus $T$ and $\beta +\mathrm{dd}^{\mathrm{c}}w$ have the same restriction to $\bigcup _ jE_ j$, and the consequence of Siu's theorem stated above identifies this restriction with $\sum _ j\nu (w,E_ j)[E_ j]$. Subtracting it from the positive current $\beta +\mathrm{dd}^{\mathrm{c}}w$ leaves the restriction of $\beta +\mathrm{dd}^{\mathrm{c}}w$ to the complement of $\bigcup _ jE_ j$, which is positive. This proves the inequality locally, hence globally. Note that $\beta +\mathrm{dd}^{\mathrm{c}}w$ itself need not be closed. $\square$
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