Theorem 6.11.4FF9DF4BE3CA8

Proof. Since the $\psi _ j$ increase to $\varphi $ almost everywhere, $\psi _ j\leq \varphi $ almost everywhere. The plurisubharmonic functions $\max \{ \psi _ j,\varphi \} $ and $\varphi $ therefore agree almost everywhere, hence everywhere by theorem 5.9.1; that is, $\psi _ j\leq \varphi $ everywhere. Consequently $\mathcal I(\psi _ j)\subset \mathcal I(\varphi )$ for every $j$, and the left side of (6.11.5) is contained in the right side. We prove the reverse inclusion by induction on the dimension $n$ of $X$ at $x$. If $n=0$, all the weights are finite at $x$ and all the ideals equal $\mathcal O_{X,x}$. Let $n\geq 1$, assume the theorem in dimension $n-1$, and take $x=0$ in a coordinate chart. We argue by contradiction, extending a germ that is missing from $\bigcup _ j\mathcal I(\psi _ j)_0$ from slices $\{ z_1=a_ i\} $ with $a_ i\to 0$, as outlined at the beginning of this section.

Step 1: reduction to a curve. The union $J=\bigcup _ j\mathcal I(\psi _ j)_0$ is an ideal, because the ideals increase with $j$. By proposition 9.2.4 it is generated by finitely many germs $g_1,\ldots ,g_ k$, $k\geq 0$, and these all belong to $\mathcal I(\psi _{j_0})_0$ for some $j_0$. Suppose, to reach a contradiction, that some $F\in \mathcal I(\varphi )_0$ does not belong to $J$; in particular $F\neq 0$. Choose representatives of $F$ and of the $g_ l$ on a neighborhood $U$ of $0$ on which $|g_ l|^2\mathrm e^{-\psi _{j_0}}$ and $|F|^2\mathrm e^{-\varphi }$ are integrable. If $|F|\leq C|g|$, with $|g|=(\sum _ l|g_ l|^2)^{1/2}$, held on a neighborhood of $0$, then $F\in \mathcal I(\psi _{j_0})_0\subset J$ by proposition 6.10.6; so no such bound holds. Lemma 6.11.1 now provides linear coordinates and a curve $\gamma (t)=(t^ q,\gamma _2(t),\ldots ,\gamma _ n(t))$ such that $F\circ \gamma $ has no zeros for small $t\neq 0$ and each $(g_ l/F)\circ \gamma $ extends holomorphically with value $0$ at $t=0$. In these coordinates choose polydiscs $\Delta _{R_0}\times P_0\Subset U$ and $\Omega =\Delta _ R\times P\Subset \Delta _{R_0}\times P_0$ centered at $0$, and let $r,c>0$ be the constants of lemma 6.11.2 for $\gamma $, $F$ and $\Omega $. Subtracting the same constant from $\varphi $ and all $\psi _ j$, we may assume that $\varphi \leq 0$ on $\Delta _{R_0}\times P_0$; then also $\psi _ j\leq 0$ there.

Step 2: choice of slices. By Fubini's theorem, the function

is integrable on $\Delta _{R_0}$. Hence, given any null set $N\subset \Delta _{R_0}$, there is a sequence of points $a_ i\notin N$, $a_ i\neq 0$, with $a_ i\to 0$ and $|a_ i|^2G(a_ i)\to 0$; otherwise $G(a)\geq c'|a|^{-2}$ for almost every $a$ in some punctured disc around $0$, and $G$ would not be integrable. We take for $N$ the union of three sets: the set where $G=\infty $; the set of $a$ for which one of the countably many functions $\varphi (a,\cdot )$, $\psi _ j(a,\cdot )$ is identically $-\infty $ on $P_0$; and the set of $a$ for which $\psi _ j(a,\cdot )\to \varphi (a,\cdot )$ fails on a subset of $P_0$ of positive measure. The first set is null because $G$ is integrable, and the other two are null by Fubini's theorem, because pole sets of weights have measure zero and $\psi _ j\to \varphi $ almost everywhere. Then the functions $\psi _ j(a_ i,\cdot )$ and $\varphi (a_ i,\cdot )$ are weights on $P_0$ (proposition 5.9.3), and the former increase to the latter almost everywhere on $P_0$. Moreover $G(a_ i)>0$ for all large $i$, because $F(a_ i,\cdot )$ does not vanish identically on $P_0$: for small $|a_ i|$, the slice $\{ a_ i\} \times P_0$ contains the points $\gamma (b)$ with $b^ q=a_ i$, at which $F\neq 0$. Discarding finitely many terms, we assume that this holds for all $i$.

Step 3: integrability on a fixed inner slice. Fix $i$. Since $G(a_ i)<\infty $, the germ of $F(a_ i,\cdot )$ at every point of $P_0$ belongs to $\mathcal I(\varphi (a_ i,\cdot ))$. By the induction hypothesis, applied on $P_0$, each point of the compact set $\overline P\subset P_0$ has a neighborhood on which $|F(a_ i,\cdot )|^2\mathrm e^{-\psi _ j(a_ i,\cdot )}$ is integrable for some $j$. Covering $\overline P$ by finitely many such neighborhoods and using that the $\psi _ j$ increase, we find one index $j$ for which $\int _ P|F(a_ i,w)|^2\mathrm e^{-\psi _ j(a_ i,w)}\, \mathrm dV_ w<\infty $. For larger $j$ these integrands decrease and converge almost everywhere to $|F(a_ i,\cdot )|^2\mathrm e^{-\varphi (a_ i,\cdot )}$, so by dominated convergence their integrals over $P$ converge to a number at most $G(a_ i)$. Since $G(a_ i)>0$, we can choose an index $j_ i$ with

The polydisc $P$ does not depend on $i$; this is what makes the estimates of the next step uniform.

Step 4: extension and contradiction. For large $i$ we have $|a_ i|<\min \{ r,R\} $, so that the normal coordinate $z_1-a_ i$ takes values in a disc of radius $2R$ on $\Omega $. The polydisc $\Omega $ is pseudoconvex by corollary 6.3.5, being a product of discs, each of which is a sublevel set of a function $|\zeta -\zeta _0|^2$ on $\mathbb C$. Applying theorem 6.5.4, after reordering the coordinates, with the normal coordinate $z_1-a_ i$, the radius $2R$ and the weight $\psi _{j_ i}$, and using the last display of Step 3, we obtain $H_ i\in \mathcal O(\Omega )$ with $H_ i(a_ i,\cdot )=F(a_ i,\cdot )$ on $P$ and

where $C_0$ is the absolute constant of that theorem. In particular the germ of $H_ i$ at $0$ lies in $\mathcal I(\psi _{j_ i})_0\subset J$, so $H_ i=\sum _ lb_ lg_ l$ near $0$ with holomorphic $b_ l$, and $(H_ i/F)\circ \gamma =\sum _ l(b_ l\circ \gamma )\cdot (g_ l/F)\circ \gamma $ extends holomorphically across $t=0$ with value $0$. Since $\psi _{j_ i}\leq 0$, the last display also bounds $\int _\Omega |H_ i|^2\, \mathrm dV$. Now lemma 6.11.2 gives $c|a_ i|^{-2}\leq 8C_0R^2G(a_ i)$ for all large $i$, contradicting $|a_ i|^2G(a_ i)\to 0$. Hence $F\in J$, which proves (6.11.5).

Finally, $\mathcal I(\varphi )_ x$ is generated by finitely many germs, which by (6.11.5) all lie in $\mathcal I(\psi _ j)_ x$ for one index $j$. Then $\mathcal I(\varphi )_ x\subset \mathcal I(\psi _ j)_ x$, and equality holds for this and every later index, since the ideals increase and are contained in $\mathcal I(\varphi )_ x$. $\square$