Lemma 5.38.9.1525C5219E69 For a quasi-plurisubharmonic function $u$ on $X$, every local divisor $E$ over $X$, and every real $m>0$, one has
\[ \max \! \left\{ \nu _ E(u)-\frac{A_ U(E)}m,0\right\} \leq \frac1m\operatorname {ord}_ E\mathcal I(mu) \leq \nu _ E(u). \]
Here $\operatorname {ord}_ E\mathcal I(mu)$ is the order along $E$ of the pulled-back ideal. Consequently, if locally $a_ m=m^{-1}\log \sum _ k|f_{m,k}|^2+O(1)$, where the $f_{m,k}$ generate $\mathcal I(mu)$, then $\nu _ E(a_ m)\to \nu _ E(u)$ as $m\to \infty $.
Proof.
The upper estimate follows from the local Bergman lower bound in proposition 5.35.1: the logarithm of the sum of squared moduli of local generators of $\mathcal I(mu)$, divided by $m$, is at least $u$ up to a locally bounded term. Pulling back and taking generic Lelong numbers along $E$ reverses this inequality.
For the lower estimate, take a local generator $f$ of the ideal and set $k=\operatorname {ord}_ E(f\circ \pi )$. Near a generic point of $E$, choose a local coordinate $z_1$ defining $E$. Divisorial removal (lemma 5.38.1), after a smooth correction, gives
\[ u\circ \pi \leq \nu _ E(u)\log |z_1|^2+C. \]
Shrink around a point where the factors of $f\circ \pi $ and of the Jacobian complementary to the powers of $z_1$ do not vanish. The change-of-variables formula and local integrability of $|f|^2\mathrm{e}^{-mu}$ then require integrability in $z_1$ of
\[ |z_1|^{2(k+A_ U(E)-1-m\nu _ E(u))}. \]
This implies $k+A_ U(E)>m\nu _ E(u)$, and in particular the weaker non-strict lower bound in the statement. Taking the minimum over a finite set of local generators and using that orders are nonnegative proves the lower estimate. Since $\nu _ E(a_ m)=m^{-1}\operatorname {ord}_ E\mathcal I(mu)$, the last assertion follows by letting $m\to \infty $.
$\square$
Pome
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