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Lemma 5.38.10548A845099FB

Lemma 5.38.10.548A845099FB Let $U$ be a complex manifold of dimension $n$, let $u$ be quasi-plurisubharmonic on $U$, and let $\pi :Y\to U$ be a modification with smooth source. Fix $y\in Y$ and an integer $m\geq 1$. Write $J_\pi $ for the holomorphic Jacobian in local coordinates near $y$. If a nonzero holomorphic germ $g$ at $\pi (y)$ belongs to $\mathcal I(mu)$, then

\[ \operatorname {ord}_ y(g\circ \pi ) \geq m\nu (u\circ \pi ,y)-n-\operatorname {ord}_ yJ_\pi . \]

Proof. If $n\geq 2$, let $\rho :\widetilde Y\to Y$ be the blowup of $y$ and let $E$ be its exceptional divisor. For $n=1$, take $\rho $ to be the identity and $E=\{ y\} $. Set $F=\pi \circ \rho $ and write $J_ F$ for its local holomorphic Jacobian determinant. Generic orders along $E$ give

\[ \nu (u\circ F,E)=\nu (u\circ \pi ,y),\qquad \operatorname {ord}_ E(g\circ F)=\operatorname {ord}_ y(g\circ \pi ). \]

On a standard blowup chart the Jacobian of $\rho $ has order $n-1$ along $E$. The chain rule therefore gives the log discrepancy over $U$ as

\[ A_ U(E)=1+\operatorname {ord}_ E J_ F =n+\operatorname {ord}_ yJ_\pi . \]

Since $g\in \mathcal I(mu)$, the order of $g\circ F$ along $E$ is at least the order of this ideal. The lower estimate in lemma 5.38.9, applied to $F$, therefore proves the inequality. $\square$

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