Lemma 6.18.1BA49EE8EFFB8

Proof. For the first assertion we pull $v$ back to log resolutions of its local presentations, where theorem 6.12.8 bounds the vanishing orders of $f$ along the components of the divisor from below. Cover $X$ by finitely many open sets $V_\kappa ^+$ on which $v$ has a presentation (6.14.6), and choose open sets $V_\kappa \Subset V_\kappa ^+$ that still cover $X$. By theorem 9.6.7, the ideal generated by the presentation on $V_\kappa ^+$ has a log resolution $\pi _\kappa \colon Y_\kappa \to V_\kappa ^+$, a sequence of blowups that is finite over $V_\kappa $; it is proper and surjective, so $\pi _\kappa ^{-1}(\overline{V_\kappa })$ and the fibers of $\pi _\kappa $ are compact. On $Y_\kappa $ we have locally $v\circ \pi _\kappa =\sum _ Ea_ E\log |s_ E|^2+O(1)$, where $\pi _\kappa ^*(g_1,\ldots ,g_ r)=\mathcal O(-\sum _ Ed_ EE)$, $E$ runs over the components of this divisor, $s_ E$ is a local equation of $E$, and $a_ E=cd_ E>0$. Let $A_ E$ be the log discrepancy of $E$ (definition 6.12.3). Only finitely many components $E$ meet the compact set $\pi _\kappa ^{-1}(\overline{V_\kappa })$, so altogether there are finitely many pairs $(a_ E,A_ E)$ with $E$ meeting one of these sets. Choose $t_0>0$ with $t_0\eta a_ E\geq A_ E$ for all of them.

Now let $x\in V_\kappa $, let $t\geq t_0$, and let $f\in \mathcal I(tv)_ x$ be defined on an open neighborhood $V'\subset V_\kappa $ of $x$. Near $x$ the function $tv$ differs by a smooth function from a plurisubharmonic function with the same multiplier ideal and of the form required in theorem 6.12.8, with coefficient $tc$. We apply that theorem over $V'$, shrunk if necessary, to the log resolution obtained by restricting $\pi _\kappa $. Each component $E'$ of the restricted divisor is a connected component of $E\cap \pi _\kappa ^{-1}(V')$ for a component $E$ as above, and it has the same coefficient $d_ E$ and the same log discrepancy $A_ E$, as we show below. Since $tc\, d_ E=ta_ E$, the theorem gives

for every component $E'$ of the restricted divisor that meets $\pi _\kappa ^{-1}(x)$; the second inequality holds because $t\geq t_0$ and $E$ meets $\pi _\kappa ^{-1}(\overline{V_\kappa })$.

The coefficient and the log discrepancy of $E'$ are those of $E$ because the vanishing orders along $E$ of a local generator of $\pi _\kappa ^*(g_1,\ldots ,g_ r)$ and of a local Jacobian determinant of $\pi _\kappa $ do not depend on the point of $E$ at which they are computed. Indeed, two local generators of this invertible ideal, like two local Jacobian determinants, differ by a unit, which has order zero along $E$; since orders add under products, they have the same order along $E$ at every point of $E$ where both are defined (lemma 6.12.1(2),(4)). Moreover, the order along $E$ of a holomorphic function is locally constant on $E_{\mathrm{reg}}$, by the last assertion of lemma 6.12.1(2), and $E_{\mathrm{reg}}$ is connected (corollary 9.3.16); here $E_{\mathrm{reg}}=E$, since the components of a simple normal crossings divisor are smooth.

If $f=0$ there is nothing to prove, so let $f\neq 0$. Shrinking $V'$, we may assume that $V'$ is connected; then $f$ does not vanish identically on any nonempty open subset of $V'$. Hence $f\circ \pi _\kappa $ does not vanish identically on any nonempty open subset of $\pi _\kappa ^{-1}(V')$, because $\pi _\kappa $ is a biholomorphism off the nowhere dense set $\pi _\kappa ^{-1}(V(g_1,\ldots ,g_ r))$; in particular, the orders of $f$ along the components of the restricted divisor are finite. Let $p\in \pi _\kappa ^{-1}(x)$, and choose coordinates $w$ on a neighborhood of $p$ in $\pi _\kappa ^{-1}(V')$ in which the components of the divisor through $p$ are $\{ w_ k=0\} $, $k=1,\ldots ,\ell $, with coefficients $a_ k=a_ E$ for the corresponding components $E$. Then $v\circ \pi _\kappa =\sum _ ka_ k\log |w_ k|^2+O(1)$ near $p$. For each $k$, let $E'_ k$ be the component of the restricted divisor that contains $\{ w_ k=0\} $ near $p$, and put $o_ k=\operatorname {ord}_{E'_ k}f$, an integer with $o_ k>t(1-\eta )a_ k$ by the inequality above. Since $f\circ \pi _\kappa $ is holomorphic on a neighborhood of $E'_ k$, its order along $E'_ k$ is the same at all points of the connected submanifold $E'_ k$, by lemma 6.12.1(3); hence $f\circ \pi _\kappa $ is divisible by $w_ k^{o_ k}$ near $p$, by lemma 6.12.1(1),(2).

By the last assertion of lemma 6.12.2, divisibility by each $w_ k^{o_ k}$ separately implies divisibility by their product: $f\circ \pi _\kappa =\prod _ kw_ k^{o_ k}\cdot h$ with $h$ holomorphic near $p$, hence bounded on a smaller neighborhood of $p$. Since $|w_ k|\leq 1$ there, we obtain $|f\circ \pi _\kappa |^2\leq C\prod _ k|w_ k|^{2t(1-\eta )a_ k}\leq C'\exp (t(1-\eta )v\circ \pi _\kappa )$. Finitely many such neighborhoods cover the compact fiber $\pi _\kappa ^{-1}(x)$, and since $\pi _\kappa $ is proper, their union contains the inverse image of a neighborhood of $x$. As $\pi _\kappa $ is surjective, the inequality $|f|^2\leq C'\exp (t(1-\eta )v)$ holds near $x$.

For the second assertion, fix a point and write $u=u_0-\sigma $ near it, with $u_0$ plurisubharmonic and $\sigma $ smooth. Then $\mathcal I(tu)=\mathcal I(tu_0)$ near the point. On a bounded polydisc $\Omega $ about the point on which $u_0$ is bounded above, theorem 6.8.9(3) provides finitely many elements of $\mathcal H_{tu_0}(\Omega )$ that generate $\mathcal I(tu_0)$ near the point, and (6.8.11) of lemma 6.8.10, applied with $m=t$ to the weight $u_0$, gives $u_0\leq t^{-1}\log \sum _ i|g_ i|^2+O(1)$ for these generators. Since $\sigma $ is bounded near the point, and any two finite systems of local generators give comparable sums of squares (the remark following definition 6.14.5), the assertion follows. $\square$