6.12 Birational change of variables and logarithmic thresholdsBC64CE221556

Proof. (1) If $h=w_1^ kh_1$ near a point $(0,w'_0)$ of $E\cap \Delta $, with $h_1$ holomorphic, then the Leibniz rule gives $\partial _{w_1}^ lh(0,w')=0$ for $l<k$ and $w'$ near $w'_0$; so $a_ l=0$ near $w'_0$, and hence on the connected set $\Delta '$ by the identity theorem. Conversely, if $a_ l=0$ for $l<k$, then $h=w_1^ k\sum _{l\geq k}a_ lw_1^{l-k}$ near $p$; the last series is obtained from the Taylor series of $h$ at $p$ by lowering the exponent of $w_1$ in each term by $k$, so it converges on the same polydiscs and defines a holomorphic function (section 9.2). If $h$ vanishes on $E$ near $p$, then $a_0=h(0,\cdot )$ vanishes near the center of $\Delta '$, hence on $\Delta '$, and the case $k=1$ shows that $h$ is divisible by $w_1$ near $p$. Applying this to $\tilde w_1$, and to $w_1$ with the roles of the two coordinate systems exchanged, we obtain $\tilde w_1=c\, w_1$ and $w_1=\tilde c\, \tilde w_1$ near $p$ with holomorphic $c$ and $\tilde c$. Then $c\tilde c=1$ off $E$; since the continuous function $c\tilde c$ equals $1$ on the dense set $\{ w_1\neq 0\} $, it equals $1$ near $p$, and $c$ is a unit.

(2) By (1), $h$ is divisible by $w_1^ k$ near $p$ if and only if it is divisible by $\tilde w_1^ k$ near $p$, so the order does not depend on the coordinates. Let $k_0$ be the smallest $l$ with $a_ l\neq 0$, and $k_0=+\infty $ if all $a_ l$ vanish. By (1), $h$ is divisible by $w_1^ k$ near $p$ if and only if $k\leq k_0$, so the order of $h$ along $E$ at $p$ is $k_0$. If all $a_ l$ vanish, then $h=0$ on $\Delta $; conversely, if $h$ vanishes near $p$, then all $a_ l$ vanish near the center of $\Delta '$, hence on $\Delta '$. The case $k_0=0$ gives the assertion about $h(0,\cdot )=a_0$, because $a_0$ vanishes identically on $\Delta '$ if and only if it vanishes near the center of $\Delta '$. Finally, let $q=(0,w'_ q)\in E\cap \Delta $. The coordinates $w$ are also coordinates at $q$ with $E=\{ w_1=0\} $ near $q$, and by their formula the coefficients of the expansion of $h$ on a small polydisc centered at $q$ are the restrictions of the same functions $a_ l$. Since $\Delta '$ is connected, $a_ l$ vanishes near $w'_ q$ if and only if it vanishes on $\Delta '$, by the identity theorem. Hence the orders of $h$ along $E$ at $q$ and at $p$ are equal.

(3) By the last assertion of (2), the order of $h$ along $E$ is a locally constant function on $E_{\mathrm{reg}}$, which is connected (corollary 9.3.16). Hence it is constant.

(4) Shrinking $\Delta $, we may assume that $g$ is defined on $\Delta $; let $b_ j$ and $e_ m$ be the coefficients of the expansions of $g$ and $hg$ on $\Delta $. By the Leibniz rule, $e_ m=\sum _{i+j=m}a_ ib_ j$. If $h$ or $g$ has infinite order, then $hg$ vanishes near $p$ and has infinite order by (2). Otherwise let $k$ and $l$ be the orders of $h$ and $g$. Then $e_ m=0$ for $m<k+l$, because each term $a_ ib_ j$ with $i+j<k+l$ has $i<k$ or $j<l$, and $e_{k+l}=a_ kb_ l$. Since $a_ k$ and $b_ l$ are not identically zero and the ring of holomorphic functions on the connected set $\Delta '$ is an integral domain, by the identity theorem, $e_{k+l}\neq 0$. By (2), the order of $hg$ is $k+l$. $\square$

Proof. Let $g=\sum _\alpha c_\alpha w^\alpha $ be the power series at $p$ of a holomorphic function, as in (5.2.1). If $g=w_ i^ ag'$ near $p$, then multiplying the series of $g'$ by $w_ i^ a$ shows that $c_\alpha =0$ whenever $\alpha _ i<a$. Conversely, if $c_\alpha =0$ whenever $\alpha _ i<a_ i$ for some $i\leq \ell $, then $g=\prod _ iw_ i^{a_ i}\cdot \sum _\alpha c_\alpha w^{\alpha -\sum _ ia_ ie_ i}$, where $e_ i$ is the $i$th unit vector, and the last series converges on the same polydiscs as that of $g$. This proves the last assertion, and it shows that $k_ i<\infty $, since otherwise all $c_\alpha $ of $h$ would vanish. Hence $h=u\prod _{i\leq \ell }w_ i^{k_ i}$ with $u$ holomorphic near $p$, and $u$ is not divisible by any $w_ i$, $i\leq \ell $, by the maximality of $k_ i$.

It remains to show that $u(p)\neq 0$. Suppose, to the contrary, that $u(p)=0$. The zero set of $u$ near $p$ lies in that of $h$, hence in $\{ w_1\cdots w_\ell =0\} $; so $\ell \geq 1$. We work in the coordinates $w$, with $p=0$. Let $P$ be the first nonzero homogeneous term of $u$; it has positive degree $m$. Choose $v\in \mathbb C^ n$ with $P(v)\neq 0$ and $v_ i\neq 0$ for all $i$, which is possible since the polynomial $P(v)v_1\cdots v_ n$ is not identically zero. Then $t\mapsto u(tv)$ has a zero of order $m$ at $t=0$; choose $\varepsilon >0$ such that it has no other zeros in $|t|\leq \varepsilon $, and such that all points $y+tv$ considered below lie in the neighborhood of $0$ where the hypothesis on the zero set holds. Since $(y,t)\mapsto u(y+tv)$ is continuous and does not vanish on the compact set $\{ 0\} \times \{ |t|=\varepsilon \} $, there is a connected neighborhood $N$ of $0$ such that $u(y+tv)\neq 0$ for $y\in N$ and $|t|=\varepsilon $. By the argument principle, the number of zeros of $t\mapsto u(y+tv)$ in $|t|<\varepsilon $, counted with multiplicity, is

a continuous integer-valued function of $y\in N$; it equals $m\geq 1$ at $y=0$, hence everywhere on $N$. So for every $y\in N$ there is $t$ with $u(y+tv)=0$, and then $y_ i+tv_ i=0$ for some $i\leq \ell $, that is, $u(\sigma _ i(y))=0$ for the linear projection $\sigma _ i(y)=y-(y_ i/v_ i)v$ onto the hyperplane $H_ i=\{ y_ i=0\} $. Hence the product of the holomorphic functions $u\circ \sigma _ i$, $i\leq \ell $, vanishes on $N$. If none of them vanished identically on $N$, each would be nonzero on a dense open subset of $N$, by the identity theorem, and so would be their product. Thus some $u\circ \sigma _ i$ vanishes on $N$; since $\sigma _ i(y)=y$ for $y\in H_ i$, the function $u$ vanishes on $H_ i\cap N$, which is a neighborhood of $0$ in $H_ i$. By the first paragraph, applied with $a=1$, $u$ is then divisible by $w_ i$ near $0$: all coefficients $c_\alpha $ of $u$ with $\alpha _ i=0$ vanish, since they are the coefficients of the restriction of $u$ to $H_ i$. This contradiction shows that $u(p)\neq 0$. $\square$

Proof. By proposition 9.8.3, $\pi $ is biholomorphic over $X\setminus S$ for an analytic subset $S\subset X$ of codimension at least two; in particular $S$ is nowhere dense. The analytic subset $\pi ^{-1}(S)$ of $Y$ is nowhere dense as well, because every nonempty open subset of $Y$ contains a nonempty open set that $\pi $ maps biholomorphically onto an open subset of $X$, and such a set is not contained in $S$. Near each of their points, nowhere dense analytic subsets lie in the zero set of a holomorphic function that does not vanish identically on a connected neighborhood (section 9.2); so $S$ and $\pi ^{-1}(S)$ are Lebesgue null sets in charts, by lemma 5.5.1.

In holomorphic coordinates the real Jacobian determinant of $\pi $, that is, the factor by which $\pi ^*$ multiplies the volume form $\mathrm dx_1\wedge \mathrm dy_1\wedge \cdots \wedge \mathrm dx_ n\wedge \mathrm dy_ n$, is $|J_\pi |^2$. Indeed, since $\frac{\mathrm i}2\mathrm dz_ j\wedge \mathrm d\bar z_ j=\mathrm dx_ j\wedge \mathrm dy_ j$, this form equals $c_ n\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n\wedge \mathrm d\bar z_1\wedge \cdots \wedge \mathrm d\bar z_ n$ with $c_ n=(\mathrm i/2)^ n(-1)^{n(n-1)/2}$, the sign coming from moving each $\mathrm dz_ j$ to the left past $\mathrm d\bar z_1,\ldots ,\mathrm d\bar z_{j-1}$, and similarly in the coordinates $w$. Since $\pi ^*(\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n)=J_\pi \, \mathrm dw_1\wedge \cdots \wedge \mathrm dw_ n$ and the conjugate identity holds for the $\mathrm d\bar z_ j$, the claim follows. The change of variables formula for nonnegative measurable functions, applied to the biholomorphism $\pi $ from $W\setminus \pi ^{-1}(S)$ onto its image, therefore gives

for every nonnegative measurable $g$ and every open set $W$ in a coordinate chart of $Y$ that is mapped into a coordinate chart of $X$, with $J_\pi $ computed in these coordinates.

Now suppose that $|f|^2\mathrm e^{-\varphi }$ is integrable on a coordinate neighborhood $V$ of $x$. Applying the formula with $g=|f|^2\mathrm e^{-\varphi }$ to small coordinate neighborhoods $W\subset \pi ^{-1}(V)$, we see that (6.12.5) is integrable near every point of $\pi ^{-1}(V)$. Conversely, suppose that (6.12.5) is integrable near every point of $\pi ^{-1}(x)$, where $f$ is defined on a coordinate neighborhood $V_0$ of $x$. Cover the compact fiber by finitely many coordinate neighborhoods $W_1,\ldots ,W_ k\subset \pi ^{-1}(V_0)$ on which it is integrable. Since $\pi $ is proper, $\pi (Y\setminus (W_1\cup \cdots \cup W_ k))$ is closed and does not contain $x$, so there is a neighborhood $V\subset V_0$ of $x$ with $\pi ^{-1}(V)\subset W_1\cup \cdots \cup W_ k$. The sets $\pi (W_ l\setminus \pi ^{-1}(S))$ cover $V\setminus S$, and $S$ is a null set, so the formula gives $\int _ V|f|^2\mathrm e^{-\varphi }\, \mathrm dV\leq \sum _ l\int _{W_ l}|f\circ \pi |^2|J_\pi |^2\mathrm e^{-\varphi \circ \pi }\, \mathrm dV_ w<\infty $.

Next let $U\subset X$ be open and $h$ a section of $\mathcal O_ Y(K_{Y/X})$ over $\pi ^{-1}(U)$; we may assume that $U$ is connected. Over $U\setminus S$ the map $\pi $ is biholomorphic and $J_\pi $ has no zeros, so $h$ is holomorphic on $\pi ^{-1}(U\setminus S)$, and $h=f\circ \pi $ there for the holomorphic function $f=h\circ (\pi |_{\pi ^{-1}(U\setminus S)})^{-1}$ on $U\setminus S$, which is connected by lemma 9.2.9. If $f\not\equiv 0$, the function $\log |f|^2$ is plurisubharmonic on $U\setminus S$ (proposition 5.10.3), so by corollary 9.3.18 it is locally bounded above near $S$; hence $f$ is locally bounded near $S$, and lemma 9.2.9 extends $f$ holomorphically to $U$. Locally $h=g/J_\pi $ with $g$ holomorphic, and $g=(f\circ \pi )J_\pi $ on the dense open set $\pi ^{-1}(U\setminus S)$, hence everywhere; so $h=f\circ \pi $. The function $f$ is unique, since it is determined on $U\setminus S$, which is dense in $U$.

Finally, let $U\subset X$ be open and $f\in \mathcal O(U)$. Since $\mathcal O_ Y(K_{Y/X})$ is locally generated by $J_\pi ^{-1}$, the function $f\circ \pi $ is a section of $\mathcal O_ Y(K_{Y/X})\mathcal I(\varphi \circ \pi )$ over $\pi ^{-1}(U)$ if and only if $(f\circ \pi )J_\pi $ is a section of $\mathcal I(\varphi \circ \pi )$ there, that is, if and only if (6.12.5) is integrable near every point of $\pi ^{-1}(U)$. By the first part, applied at every point of $U$, this holds if and only if $f$ is a section of $\mathcal I(\varphi )$ over $U$. Together with the second part, which shows that every section of the right side of (6.12.6) over $\pi ^{-1}(U)$ has the form $f\circ \pi $, this proves (6.12.6). $\square$

Proof. Let $p\in \pi ^{-1}(V)$, and choose coordinates $w$ at $p$ as before (6.12.7), so that $E_1,\ldots ,E_\ell $ are the components of $D$ through $p$ and $E_ i=\{ w_ i=0\} $ near $p$. Since $b$ is bounded, (6.12.7), applied to $\pi \colon Y\to V$, gives

near $p$.

Let $h$ be holomorphic near $p$; we decide when $|h|^2\prod _{i\leq \ell }|w_ i|^{-2cd_ i}$, or equivalently $|h|^2\mathrm e^{-\varphi \circ \pi }$, is integrable near $p$. Let $k_ i$ be the order of $h$ along $E_ i$ at $p$, which is finite if $h$ does not vanish identically near $p$; if it does, the integrability is trivial and $k_ i=+\infty $. We claim that the integrability holds if and only if $k_ i+1>cd_ i$ for $i=1,\ldots ,\ell $. If these inequalities hold, write $h=\prod _ iw_ i^{k_ i}\tilde h$ with $\tilde h$ holomorphic, by lemma 6.12.2; the integrand is at most a constant times $\prod _ i|w_ i|^{2k_ i-2cd_ i}$, which is integrable near $p$ by Fubini's theorem and polar coordinates in each $w_ i$, since $2k_ i-2cd_ i>-2$. Conversely, suppose that $k_1+1\leq cd_1$, say, and write $h=w_1^{k_1}\tilde h$ with $\tilde h$ not divisible by $w_1$. Then $\tilde h$ does not vanish identically on $\{ w_1=0\} $ near $p$, by lemma 6.12.1(2), so arbitrarily close to $p$ there are an open set $W'$ of points $w'=(w_2,\ldots ,w_ n)$ and $\delta ,\rho >0$ with $|\tilde h(w_1,w')|\geq \delta $ for $|w_1|<\rho $ and $w'\in W'$. There the integrand is at least a positive constant times $|w_1|^{2k_1-2cd_1}$, since the other factors $|w_ i|^{-2cd_ i}$ are bounded below; as $2k_1-2cd_1\leq -2$, its integral in $w_1$ diverges, and by Fubini's theorem the integrand is not integrable near $p$.

Now let $f$ be holomorphic on $V$, and apply this to $h=(f\circ \pi )J_\pi $, with $J_\pi $ computed in local coordinates near $p$. The function $f\circ \pi $ is holomorphic on $\pi ^{-1}(V)$, which contains $E_ i$, so its order along $E_ i$ at $p$ is $\operatorname {ord}_{E_ i}f$, by lemma 6.12.1(3) and definition 6.12.3; the order of $J_\pi $ along $E_ i$ at $p$ is $\operatorname {ord}_{E_ i}J_\pi $. Since orders add under products (lemma 6.12.1(4)), $k_ i=\operatorname {ord}_{E_ i}f+\operatorname {ord}_{E_ i}J_\pi =\operatorname {ord}_{E_ i}f+A_ X(E_ i)-1$. Thus, for $p\in \pi ^{-1}(x)$, the condition at $p$ is (6.12.9) for the components through $p$. As $p$ ranges over $\pi ^{-1}(x)$, these are exactly the components meeting the fiber, and proposition 6.12.4 proves the first assertion.

For (6.12.10), let $U\subset V$ be open and $f\in \mathcal O(U)$, and put $h=(f\circ \pi )J_\pi $ on $\pi ^{-1}(U)$. For an integer $k$, the strict inequality $k+1>cd_ i$ is equivalent to $k\geq \lfloor cd_ i\rfloor $. By proposition 6.12.4 and the claim, applied at every point $p\in \pi ^{-1}(U)$, we have $f\in \mathcal I(\varphi )(U)$ if and only if at every such $p$ the order of $h$ at $p$ along each component $E_ i$ through $p$ is at least $\lfloor cd_ i\rfloor $. By lemma 6.12.2, this means that $h$ is divisible near every point of $\pi ^{-1}(U)$ by the local equation $\prod _ iw_ i^{\lfloor cd_ i\rfloor }$ of $\lfloor cD\rfloor $, that is, that $f\circ \pi =h/J_\pi $ is a section of $\mathcal O_ Y(K_{Y/X}-\lfloor cD\rfloor )$ over $\pi ^{-1}(U)$. Since $\lfloor cD\rfloor $ is effective, every section of $\mathcal O_ Y(K_{Y/X}-\lfloor cD\rfloor )$ over $\pi ^{-1}(U)$ is a section of $\mathcal O_ Y(K_{Y/X})$, and hence of the form $f\circ \pi $ with $f\in \mathcal O(U)$, by proposition 6.12.4. This proves (6.12.10) on $V$. $\square$

Proof. We may assume $k<\infty $. Choose coordinates $w=(w_1,w')$ at $p$ with $E=\{ w_1=0\} $, and write $h=w_1^ kv$ with $v$ holomorphic and not divisible by $w_1$; then $v(0,\cdot )$ does not vanish identically near $p$, by lemma 6.12.1(2). Moving $p$ slightly along $E$, we may assume that $v(p)\neq 0$, so that $|v|\geq \delta >0$ on a neighborhood $\{ |w_1|<\rho \} \times W'$ of $p$ on which $|h|^2\mathrm e^{-\psi }$ is integrable. By Fubini's theorem, for almost every $w'\in W'$ the function $w_1\mapsto |w_1|^{2k}|v|^2\mathrm e^{-\psi (w_1,w')}$ is integrable on $|w_1|<\rho $; moreover, for almost every $w'$ the function $u=\psi (\cdot ,w')$ is not identically $-\infty $ on this disc, since the pole set of $\psi $ is a null set.

Fix such a $w'$. By proposition 5.9.3, $u$ is subharmonic on the disc, and its Lelong number at $0$ is at least $\nu (\psi ,(0,w'))$: restricting the bound $\psi (w)\leq c\log \bigl (|w-(0,w')|^2/r_0^2\bigr )+C$ of (6.7.3), with $c=\nu (\psi ,(0,w'))$, to the line through $(0,w')$ in the $w_1$-direction gives $u(w_1)\leq c\log (|w_1|^2/r_0^2)+C$. Moreover $\nu (\psi ,(0,w'))\geq \nu (\psi ,E)$, the latter being an infimum over $E$. Hence, by (6.7.3) for $u$, there are $r_1\in (0,\rho ]$ and $c'>0$ with $\mathrm e^{-u(w_1)}\geq c'|w_1|^{-2\nu (\psi ,E)}$ for $0<|w_1|<r_1$. Therefore

which forces $2k-2\nu (\psi ,E)+2>0$, that is, $k+1>\nu (\psi ,E)$. $\square$

Proof. By corollary 6.11.6 there are $\varepsilon >0$ and a neighborhood $V$ of $x$ on which $f$ is defined such that $|f|^2\mathrm e^{-(1+\varepsilon )\varphi }$ is integrable on $V$. Let $E$ be a prime divisor over an open set $U\subset V$, represented on a smooth model $\pi \colon Y\to U$; we may assume that $\operatorname {ord}_ Ef<\infty $. Since $f\in \mathcal I((1+\varepsilon )\varphi )_ y$ for every $y\in U$, proposition 6.12.4, applied to the weight $(1+\varepsilon )\varphi $ on $U$, shows that $|h|^2\mathrm e^{-(1+\varepsilon )\varphi \circ \pi }$, with $h=(f\circ \pi )J_\pi $, is integrable near every point of $Y$. Let $p\in E_{\mathrm{reg}}$. Since orders add under products, the order of $h$ along $E$ at $p$ is $\operatorname {ord}_ Ef+\operatorname {ord}_ EJ_\pi =\operatorname {ord}_ Ef+A_ X(E)-1$. By lemma 6.12.11, applied to the weight $(1+\varepsilon )\varphi \circ \pi $, whose generic Lelong number along $E$ is $(1+\varepsilon )\nu (\varphi ,E)$ by proposition 6.7.9, we obtain (6.12.13). $\square$

Proof. Let $k=\operatorname {ord}_ E\mathcal I(m\varphi )$, and choose a point $p\in E_{\mathrm{reg}}$. Put $y=\pi (p)$, choose a bounded polydisc $\Omega $ around $y$ in a chart on which $\varphi $ is bounded above, and choose $g_1,\ldots ,g_ r\in \mathcal H_{m\varphi }(\Omega )$ generating $\mathcal I(m\varphi )$ near $y$ (theorem 6.8.9(3)). Near $p$, in coordinates $w=(w_1,w')$ with $E=\{ w_1=0\} $, let $k_ i$ be the order of $g_ i\circ \pi $ along $E$ at $p$, so that $k=\min _ ik_ i$; since $\mathcal I(m\varphi )_ y\neq 0$, this minimum is finite.

For the lower bound, choose $i$ with $k_ i=k$. Since $|g_ i|^2\mathrm e^{-m\varphi }$ is integrable on $\Omega $, proposition 6.12.4, applied to the weight $m\varphi $, shows that $|h|^2\mathrm e^{-m\varphi \circ \pi }$, with $h=(g_ i\circ \pi )J_\pi $, is integrable near $p$. The order of $h$ along $E$ at $p$ is $k+A_ X(E)-1$, because orders add under products. By lemma 6.12.11, applied to the weight $m\varphi \circ \pi $, whose generic Lelong number along $E$ is $m\nu (\varphi ,E)$ by proposition 6.7.9, we obtain $k+A_ X(E)>m\nu (\varphi ,E)$. This is the left inequality.

For the upper bound, lemma 6.8.10 gives $m\varphi \leq \log \sum _ i|g_ i|^2+mC_ m$ on a neighborhood of $y$. Near $p$ we have $g_ i\circ \pi =w_1^{k_ i}u_ i$ with $u_ i$ holomorphic, so $|g_ i\circ \pi |\leq C|w_1|^{k_ i}\leq C|w_1|^ k$ where $|w_1|\leq 1$; it follows that $m\varphi \circ \pi \leq k\log |w_1|^2+C''$ on a neighborhood $N$ of $p$. For $q\in E\cap N$ we have $|w_1|\leq |w-q|$, so $m\varphi \circ \pi (w)\leq k\log |w-q|^2+C''$ near $q$, and proposition 6.7.1, proposition 6.7.9 give $\nu (\varphi \circ \pi ,q)\geq k/m$. Since the generic value $\nu (\varphi \circ \pi ,E)$ is attained at points of $E$ arbitrarily close to $p$ (proposition 6.9.7), we conclude that $m\nu (\varphi ,E)\geq k$. Dividing (6.12.15) by $m$ and letting $m\to \infty $ gives the limit formula. $\square$

Proof. Necessity, even with strict inequality, is proposition 6.12.12. Since the strict condition implies the non-strict one, it suffices to show that the non-strict condition implies $f\in \mathcal I(\varphi )_ x$; we may assume $f\neq 0$. We compare $\varphi $ with a weight $\varphi _ m$ with analytic singularities defined by generators of $\mathcal I(m\varphi )$, apply theorem 6.12.8 to a multiple of $\varphi _ m$, and then return to $\varphi $. Put $\lambda =1+\varepsilon /2$ and choose an integer $m\geq 2$ with $\lambda \geq m/(m-1)$, that is, $(m-1)\lambda \geq m$. On a bounded polydisc $\Omega \subset V$ around $x$ on which $\varphi $ is bounded above, choose $g_1,\ldots ,g_ r\in \mathcal H_{m\varphi }(\Omega )$ generating $\mathcal I(m\varphi )$ near $x$, and put

This is plurisubharmonic by proposition 5.10.3, since $\mathcal I(m\varphi )_ x\neq 0$ by lemma 6.9.1, so that the $g_ i$ do not all vanish identically. By (6.8.11), $\varphi \leq \varphi _ m+C_ m$ on a neighborhood $W\subset V$ of $x$. Pulling this back to a smooth model and using the characterization of Lelong numbers in proposition 6.7.1 at every point, we see that $\nu (\varphi _ m,E)\leq \nu (\varphi ,E)$ for every prime divisor $E$ over an open subset of $W$. Hence, for every prime divisor $E$ over an open neighborhood of $x$ contained in $W$ whose center contains $x$,

and this is strictly larger than $\lambda \nu (\varphi _ m,E)$: when $\nu (\varphi _ m,E)>0$ because $1+\varepsilon >\lambda $, and when $\nu (\varphi _ m,E)=0$ because the left side is at least $A_ X(E)\geq 1$.

Now let $\pi \colon Y\to W'$ be a log resolution of the ideal $(g_1,\ldots ,g_ r)$ over an open neighborhood $W'\subset W$ of $x$, with $(g_1,\ldots ,g_ r)\mathcal O_ Y=\mathcal O_ Y(-D)$ and $D=\sum _ id_ iE_ i$. Let $q\in E_ i$, and choose coordinates $w$ at $q$ as before (6.12.7), so that the components of $D$ through $q$ are coordinate hyperplanes and $E_ i=\{ w_ i=0\} $. By (6.12.7), $\log \sum _ l|g_ l\circ \pi |^2=\sum _ jd_ j\log |w_ j|^2+O(1)\leq d_ i\log |w_ i|^2+C$ near $q$, where $j$ runs over the components through $q$; since $|w_ i|\leq |w-q|$, the characterization in proposition 6.7.1 gives $\nu (\varphi _ m\circ \pi ,q)\geq d_ i/m$. If $q$ lies on no other component of $D$, then $\varphi _ m\circ \pi =(d_ i/m)\log |w_ i|^2+O(1)$ near $q$, and $\nu (\varphi _ m\circ \pi ,q)=d_ i/m$ by proposition 6.7.9. Such points exist, because $E_ i$ is not contained in the union of the other components of the support of $D$ (corollary 9.3.16). Hence $\nu (\varphi _ m,E_ i)=d_ i/m$. Every component $E_ i$ meeting $\pi ^{-1}(x)$ has a center containing $x$, so the strict inequality above reads $\operatorname {ord}_{E_ i}f+A_ X(E_ i)>(\lambda /m)d_ i$ for these components, and theorem 6.12.8, applied on $W'$ to the weight $\lambda \varphi _ m$ with $c=\lambda /m$ and $b=0$, gives $f\in \mathcal I(\lambda \varphi _ m)_ x$.

It remains to pass from $\lambda \varphi _ m$ to $\varphi $. By the definition of the generators,

near $x$. Shrinking the neighborhood and subtracting a constant from $\varphi _ m$, we may assume $\varphi _ m\leq 0$; this multiplies the functions in question by constants and changes none of the integrability statements. We split a small neighborhood of $x$ into the set where $\varphi \geq \lambda \varphi _ m$ and its complement, ignoring the null set formed by the poles of $\varphi $ and the common zeros of the $g_ i$, so that all exponents below are finite. On the first set, $|f|^2\mathrm e^{-\varphi }\leq |f|^2\mathrm e^{-\lambda \varphi _ m}$, which is integrable because $f\in \mathcal I(\lambda \varphi _ m)_ x$. On the complement we have $(m-1)\varphi <(m-1)\lambda \varphi _ m$, since $m\geq 2$; as $(m-1)\lambda \geq m$ and $\varphi _ m\leq 0$, it follows that

and therefore

where $C$ is a local bound for $|f|^2$. By (6.12.18) the right side is integrable near $x$. Hence $|f|^2\mathrm e^{-\varphi }$ is integrable near $x$, that is, $f\in \mathcal I(\varphi )_ x$. $\square$