Corollary 5.43.26.50D99B4B0660 Let $\theta $ be a smooth real $(1,1)$-form and $u\in \mathrm{PSH}(X,\theta )$, and let $u_ m$, with curvature errors $\epsilon _ m$, be the approximants of corollary 5.36.2. Let $Y\subset X$ be a connected compact complex submanifold along which the generic Lelong number of $u$ vanishes:
\[ \nu (u,Y)=\inf _{y\in Y}\nu (u,y)=0. \]
Then each restriction $u_ m|_ Y$ is quasi-plurisubharmonic on $Y$; in particular, it is not identically $-\infty $. If $\pi :Z\to Y$ is a modification with smooth compact source and $E\subset Z$ is a prime divisor, then
\[ \sup _ m\nu \bigl ((u_ m|_ Y)\circ \pi ,E\bigr )<\infty . \]
Proof.
For each $m$, choose $y\in Y$ with $\nu (u,y)<1/m$. Skoda's local integrability criterion gives $\mathcal I(mu)_ y=\mathcal O_{X,y}$, so the image ideal $\mathcal I(mu)\mathcal O_ Y$ does not vanish identically. Since $u_ m$ has analytic type $(1/m,\mathcal I(mu))$, proposition 5.34.1 shows that $u_ m|_ Y$ is quasi-plurisubharmonic on $Y$. Its pullback to $Z$ is quasi-plurisubharmonic as well, since $\pi $ is biholomorphic off a proper analytic subset.
Restricting and pulling back the curvature inequality of corollary 5.36.2, which is allowed by lemma 5.43.2, gives
\[ \pi ^*(\theta |_ Y)+\mathrm{dd}^{\mathrm{c}}\bigl ((u_ m|_ Y)\circ \pi \bigr ) \geq -\epsilon _ m\pi ^*(\omega |_ Y). \]
Since the errors $\epsilon _ m$ are bounded, a sufficiently large multiple of a single Hermitian form on $Z$ dominates $\pi ^*(\theta |_ Y)+\epsilon _ m\pi ^*(\omega |_ Y)$ for all $m$. Lemma 5.38.8, applied on $Z$ in this fixed background, therefore bounds the generic Lelong numbers along $E$ uniformly in $m$.
$\square$
Pome
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