6.10 Restriction, products and integral closure3E60ECFFA1E9

Proof. The inclusion (6.10.2) is a restatement of corollary 6.5.9: if $f\in \mathcal I(\varphi |_ Z)_ x$, that corollary provides a holomorphic germ $F$ at $x$ with $F|_ Z=f$ and $F\in \mathcal I(\varphi )_ x$.

The product formula is local. Near a point $(x_1,x_2)$ we choose bounded polydiscs $\Omega _ i$ around $x_ i$ in charts of $X_ i$ on which $\varphi _ i$ is bounded above; subtracting constants, which changes none of the multiplier ideals, we may assume that $\varphi _1,\varphi _2\leq 0$. We write $\Phi =\varphi _1\circ p_1+\varphi _2\circ p_2$, a nonpositive weight on the polydisc $\Omega _1\times \Omega _2$. Let $(e_ j)$ and $(g_ k)$ be orthonormal bases of $\mathcal H_{\varphi _1}(\Omega _1)$ and $\mathcal H_{\varphi _2}(\Omega _2)$, which exist by theorem 6.8.9(2). By Fubini's theorem the functions $e_ j(z)g_ k(w)$ form an orthonormal family in $\mathcal H_\Phi (\Omega _1\times \Omega _2)$. We show that its linear span is dense. Let $F\in \mathcal H_\Phi (\Omega _1\times \Omega _2)$. By Fubini's theorem, for almost every $w\in \Omega _2$ we have $\varphi _2(w)>-\infty $ and $F(\cdot ,w)\in \mathcal H_{\varphi _1}(\Omega _1)$, and for such $w$ we put

We claim that each $a_ j$ agrees almost everywhere with an element of $\mathcal H_{\varphi _2}(\Omega _2)$ of norm at most $\| F\| _\Phi $. Choose compact sets $K_\ell \uparrow \Omega _1$ and put $E_\ell =K_\ell \cap \{ \mathrm e^{-\varphi _1}\leq \ell \} $; these sets increase and exhaust $\Omega _1$ up to the pole set of $\varphi _1$, which has measure zero. The function $a_{j,\ell }$ obtained by integrating over $E_\ell $ instead of $\Omega _1$ is holomorphic on $\Omega _2$. Indeed, let $L\subset \Omega _2$ be compact. For $z\in E_\ell \subset K_\ell $ and $w\in L$, the integrand $F(z,w)\overline{e_ j(z)}\mathrm e^{-\varphi _1(z)}$ and all its partial derivatives in $w$ are bounded, because $F$ and its derivatives are continuous on $K_\ell \times L$, $e_ j$ is bounded on $K_\ell $, and $\mathrm e^{-\varphi _1}\leq \ell $ on $E_\ell $. Hence we may differentiate under the integral sign any number of times: $a_{j,\ell }$ is smooth, and $\partial a_{j,\ell }/\partial \bar w_ m=0$ for all $m$, because $F$ is holomorphic in $w$. By the Cauchy–Schwarz inequality in $z$, followed by integration in $w$ against $\mathrm e^{-\varphi _2}$,

by dominated convergence, since the sets $\Omega _1\setminus E_\ell $ decrease to a null set and $|e_ j|^2\mathrm e^{-\varphi _1}$ is integrable. Since $\varphi _2\leq 0$, the holomorphic functions $a_{j,\ell }$ therefore converge to $a_ j$ in $L^2(\Omega _2)$, and hence in the sense of distributions; so $\bar\partial a_ j=0$, and $a_ j$ agrees almost everywhere with a holomorphic function, by lemma 6.4.10(1). The same Cauchy–Schwarz estimate over all of $\Omega _1$ gives $\| a_ j\| _{\varphi _2}\leq \| F\| _\Phi $, which proves the claim.

Now suppose that $F$ is orthogonal to all the products $e_ jg_ k$. By Fubini's theorem, whose use is justified by the Cauchy–Schwarz inequality, $\langle a_ j,g_ k\rangle _{\varphi _2}=\langle F,e_ jg_ k\rangle _\Phi =0$ for all $k$. Since the linear span of the $g_ k$ is dense in $\mathcal H_{\varphi _2}(\Omega _2)$, the element $a_ j$ of this space is orthogonal to itself, so $a_ j=0$ almost everywhere, for every $j$. Hence for almost every $w$ all the coefficients $\langle F(\cdot ,w),e_ j\rangle _{\varphi _1}$ vanish, and $F(\cdot ,w)=0$, because $F(\cdot ,w)$ is the limit of the partial sums of its expansion (theorem 6.8.9(2)). Thus $F=0$ almost everywhere, and therefore everywhere, since $F$ is continuous. Consequently, the closure $M$ of the linear span of the products $e_ jg_ k$ satisfies $M^\perp =0$, and $M=\mathcal H_\Phi (\Omega _1\times \Omega _2)$ by proposition 6.2.1.

We can now prove (6.10.3). Denote its right side by $\mathfrak b$; it is the ideal generated by the products $(f_1\circ p_1)(f_2\circ p_2)$ of local sections $f_ i$ of $\mathcal I(\varphi _ i)$. Each product $e_ jg_ k$ is a section of $\mathfrak b$. Every $F\in \mathcal H_\Phi (\Omega _1\times \Omega _2)$ is the norm limit, hence by (6.8.7) the locally uniform limit, of a sequence of finite linear combinations of these products, so its germs lie in $\mathfrak b$ by lemma 6.8.1. Since $\mathcal I(\Phi )$ is generated by the germs of the elements of $\mathcal H_\Phi (\Omega _1\times \Omega _2)$ (theorem 6.8.9(3)), we obtain $\mathcal I(\Phi )\subset \mathfrak b$. Conversely, if $|f_ i|^2\mathrm e^{-\varphi _ i}$ is integrable near $x_ i$ for $i=1,2$, then $|f_1(z)f_2(w)|^2\mathrm e^{-\Phi (z,w)}$ is integrable near $(x_1,x_2)$ by Fubini's theorem; since $\mathcal I(\Phi )$ is an ideal, $\mathfrak b\subset \mathcal I(\Phi )$.

Finally, let $\varphi $ and $\psi $ be weights on $X$, and apply (6.10.2) to the weight $\varphi \circ p_1+\psi \circ p_2$ on $X\times X$ and the diagonal, which we identify with $X$. The restriction of the weight is $\varphi +\psi $, and by (6.10.3) the restriction of its multiplier ideal is generated by the products $f_1f_2$ of local sections $f_1$ of $\mathcal I(\varphi )$ and $f_2$ of $\mathcal I(\psi )$, that is, it equals $\mathcal I(\varphi )\mathcal I(\psi )$. This proves (6.10.4). $\square$

Proof. Suppose first that $f$ is a submersion, and let $a\in X$. Choose holomorphic coordinates $y$ on $Y$ centered at $f(a)$. Since the differential of $f$ at $a$ is surjective, the complex Jacobian matrix at $a$ of the components of $y\circ f$, computed in a chart of $X$, has rank $\dim _{f(a)}Y$. By the holomorphic inverse function theorem (proposition 5.2.4(3)), these components are the first coordinates of a holomorphic chart $(z,w)=(y\circ f,w)$ of $X$ near $a$. In these coordinates $f$ is the projection $(z,w)\mapsto z$. Thus $\varphi \circ f=\varphi \circ p_1+0\circ p_2$, and (6.10.3), together with $\mathcal I(0)=\mathcal O$, gives $\mathcal I(\varphi \circ f)=p_1^{-1}\mathcal I(\varphi )\cdot \mathcal O=f^{-1}\mathcal I(\varphi )\cdot \mathcal O_ X$.

In general, the graph $\Gamma =\{ (x,f(x))\} $ is a complex submanifold of $X\times Y$, since in local coordinates $(x,y)$ the functions $(x,y-f(x))$ are holomorphic coordinates in which $\Gamma =\{ y-f(x)=0\} $; and $\iota (x)=(x,f(x))$ is a biholomorphism of $X$ onto $\Gamma $. The projection $q\colon X\times Y\to Y$ is a submersion, so $\mathcal I(\varphi \circ q)=q^{-1}\mathcal I(\varphi )\cdot \mathcal O_{X\times Y}$ by the first case. Since $\varphi \circ q\circ \iota =\varphi \circ f$, (6.10.2) applied to $\Gamma $ gives $\mathcal I(\varphi \circ f)\subset (q^{-1}\mathcal I(\varphi )\cdot \mathcal O_{X\times Y})|_\Gamma ^{\mathrm{im}}$ after transport by $\iota $. The right side is generated by the functions $g\circ q\circ \iota =g\circ f$ with $g$ a local section of $\mathcal I(\varphi )$, so it equals $f^{-1}\mathcal I(\varphi )\cdot \mathcal O_ X$. $\square$

Proof. The second assertion is immediate, since $|F|^2\mathrm e^{-\varphi }\leq C^2\sum _ j|g_ j|^2\mathrm e^{-\varphi }$ near $x$. For the first, choose generators $g_1,\ldots ,g_ r$ of $\mathcal I(\varphi )_ x$ (theorem 6.8.9), representatives of the $g_ j$, of $F$ and of the $a_ j$ on a common neighborhood, and put $|g|=(\sum _ j|g_ j|^2)^{1/2}$. Each $a_ j$ is a finite sum of products of $j$ generators with holomorphic coefficients, which are bounded on a smaller neighborhood; hence $|a_ j|\leq C_ j|g|^ j$ there. At a point where $|g|>0$, put $\lambda =|F|/|g|$. Dividing the equation by $|g|^ d$ gives $\lambda ^ d\leq \sum _ jC_ j\lambda ^{d-j}$, which forces $\lambda \leq C$ with $C=\max \{ 1,\sum _ jC_ j\} $. At a point where $|g|=0$, all the $a_ j$ vanish, so $F^ d=0$ and $F=0$. Thus $|F|\leq C|g|$ near $x$, and the second assertion applies. $\square$