Lemma 18.27.296F4F5465740

Proof. The function $q_\delta /\delta $ belongs to $\mathrm{PSH}(X,h)$. The uniform Lelong bound of lemma 5.38.8, and invariance under bounded changes, give

Letting $\delta $ tend to zero proves the first assertion.

For polar curvature, put $S=\{ q=-\infty \} $ and let $R=\mathbf1_ S\mathrm{dd}^{\mathrm{c}}q$, a closed positive current by proposition 5.46.7. Choose a Gauduchon metric $\gamma $ when $n\geq 2$, using theorem 11.4.2; when $n=1$ take any Hermitian form and use $\gamma ^0=1$. Bounded-type invariance in proposition 5.46.3 gives $R=\mathbf1_ S(\delta h+\mathrm{dd}^{\mathrm{c}}q_\delta )$. The positive current $\delta h+\mathrm{dd}^{\mathrm{c}}q_\delta $ has total $\gamma $-trace $\delta \int _ Xh\wedge \gamma ^{n-1}$ by distributional Stokes. Consequently

Thus $R=0$ by positivity and $\delta \downarrow 0$.

Now let $\pi :X'\to X$ be a smooth connected compact modification, and fix $h'>0$ on $X'$. Genuine pullbacks are quasi-plurisubharmonic by proposition 5.43.18. Their ordinary inequalities hold pointwise, including at poles, and $\pi ^*h\leq Ch'$ for some fixed $C$. Hence $\pi ^*q_\delta \sim \pi ^*q$ and $\mathrm{dd}^{\mathrm{c}}\pi ^*q_\delta \geq -C\delta h'$. Since $C\delta $ can be arbitrarily small, the preceding argument on $X'$ proves both vanishing assertions. $\square$