6.11 Strong openness by extension from moving slicesF441BF47EB53

Proof. We first construct a nonconstant holomorphic map $\delta $ from a disc around $0\in \mathbb C$ to $\mathbb C^ n$ with $\delta (0)=0$ such that $F\circ \delta $ has no zeros other than $\tau =0$ and each $(g_ j/F)\circ \delta $ extends holomorphically across $\tau =0$ with value $0$.

If $r=0$, choose a vector $v$ such that $\zeta \mapsto F(\zeta v)$ does not vanish identically near $0$; for instance, $v$ may be any vector at which the first nonzero homogeneous term of $F$ at $0$ does not vanish. The zeros of this one-variable function are isolated, so on a small disc it vanishes at most at $0$, and $\delta (\tau )=\tau v$ has the required properties.

Let $r\geq 1$. The bound $|F|\leq i|g|$ fails on the ball $B(0,1/i)$ for every large integer $i$, so there are points $z_ i$ with $|z_ i|<1/i$ and $|F(z_ i)|>i|g(z_ i)|$; in particular $F(z_ i)\neq 0$ and $|g(z_ i)|/|F(z_ i)|<1/i$. We may take $z_ i\neq 0$: if $z_ i=0$, the strict inequality $|F|>i|g|$ at $0$ persists at points $z\neq 0$ near $0$, because $F$ and $|g|$ are continuous.

Let $U$ be a connected neighborhood of $0$ on which all functions are defined, and let $\sigma \colon \widetilde U\to U$ be the blowup of the ideal $(F,g_1,\ldots ,g_ r)$, which is not identically zero on $U$ (definition 9.6.1). Thus $\widetilde U\subset U\times \mathbb P^ r$ is the closure of the graph of $z\mapsto [F(z):g_1(z):\cdots :g_ r(z)]$ over the complement of the common zeros, and it is a complex space by proposition 9.6.2. The graph points $\tilde z_ i=(z_ i,[F(z_ i):g(z_ i)])$ converge to $b=(0,[1:0:\cdots :0])$, because $g(z_ i)/F(z_ i)\to 0$. On the part of $\widetilde U$ lying over the chart $\{ \xi _0\neq 0\} $ of $\mathbb P^ r$, the functions $\xi _ j/\xi _0$ are holomorphic and vanish at $b$, and the identities $g_ j\circ \sigma =(\xi _ j/\xi _0)(F\circ \sigma )$ hold on the graph, hence, both sides being continuous, on its closure, which is this part of $\widetilde U$.

The germ of $\widetilde U$ at $b$ is the union of finitely many irreducible branches (proposition 9.3.1); passing to a subsequence, we may assume that all $\tilde z_ i$ lie on a representative $W$ of one branch. The analytic subgerm of $(W,b)$ formed by $\sigma ^{-1}(0)\cup \{ F\circ \sigma =0\} $ is different from $(W,b)$, because $W$ contains the points $\tilde z_ i$, arbitrarily close to $b$, at which $z_ i\neq 0$ and $F(z_ i)\neq 0$; for the same reason $(W,b)$ is not the point $b$ and has positive dimension. By proposition 9.5.2 there is a nonconstant holomorphic map $c$ from a disc into $W$ with $c(0)=b$ such that $c(\tau )\notin \sigma ^{-1}(0)$ and $F(\sigma (c(\tau )))\neq 0$ for $\tau \neq 0$; shrinking the disc, we may assume that $c$ takes values over the chart $\{ \xi _0\neq 0\} $. Put $\delta =\sigma \circ c$. Then $\delta (0)=0$, and for $\tau \neq 0$ we have $\delta (\tau )\neq 0$ and $F(\delta (\tau ))\neq 0$; so $\delta $ is not constant, and $F\circ \delta $ has no zeros other than $\tau =0$. For $\tau \neq 0$ the identities above give $(g_ j/F)(\delta (\tau ))=(\xi _ j/\xi _0)(c(\tau ))$, and the right side is holomorphic on the whole disc with value $(\xi _ j/\xi _0)(b)=0$ at $\tau =0$.

Finally we choose the coordinates and the parameter. Let $q$ be the smallest vanishing order at $0$ of the coordinate functions of $\delta $; it is finite because $\delta $ is not constant, and $q\geq 1$ because $\delta (0)=0$. Then $\delta (\tau )=\tau ^ qe+O(\tau ^{q+1})$ with a vector $e\neq 0$. Choose a linear form $\ell $ with $\ell (e)\neq 0$ and linear coordinates in which $\ell $ is the first coordinate; then $\delta _1(\tau )=\tau ^ qu(\tau )$ with $u$ holomorphic and $u(0)=\ell (e)\neq 0$, while the other coordinates satisfy $|\delta _ j(\tau )|\leq C|\tau |^ q$ near $0$. On a disc around $0$ on which $u$ has no zeros, $u$ has a holomorphic logarithm $h$, and $\kappa =\mathrm e^{h/q}$ is a holomorphic $q$th root of $u$. The function $s(\tau )=\tau \kappa (\tau )$ satisfies $s'(0)=\kappa (0)\neq 0$, so by the holomorphic inverse function theorem in one variable (proposition 5.2.4(1) with $n=1$) it maps a neighborhood of $0$ biholomorphically onto a disc around $0$. Its inverse $s^{-1}$ is holomorphic and vanishes at $0$, so that $|s^{-1}(t)|\leq C'|t|$ after shrinking the disc. Put $\gamma =\delta \circ s^{-1}$ on this disc. Then $\gamma _1(t)=s^{-1}(t)^ qu(s^{-1}(t))=s(s^{-1}(t))^ q=t^ q$, and $|\gamma _ j(t)|\leq CC'^ q|t|^ q$ for $j\geq 2$. Since $s^{-1}$ maps the punctured disc onto a punctured neighborhood of $0$, $F\circ \gamma $ has no zeros other than $t=0$, and each $(g_ j/F)\circ \gamma =((g_ j/F)\circ \delta )\circ s^{-1}$ extends holomorphically across $t=0$ with value $0$. $\square$

Proof. We bound the norm of $H$ from below by a weighted norm of $H\circ \gamma $ on a disc, and then use that $H\circ \gamma =F\circ \gamma $ at the points $t$ with $t^ q=a$.

Choose $\rho >0$ with $\rho ^ q<R$ such that $\gamma $ is defined on a neighborhood of the closed disc $|t|\leq \rho $, $F\circ \gamma $ has no zeros for $0<|t|\leq \rho $, and, since $\gamma $ is continuous with $\gamma (0)=0$, $(\gamma _2(t),\ldots ,\gamma _ n(t))$ lies in a fixed polydisc $P'\Subset P$ centered at $0$ for $|t|\leq \rho $. Let $\delta >0$ be half the distance from $P'$ to the complement of $P$. For $0<|z_1|<\rho ^ q$, each of the $q$ points $\gamma (t)$ with $t^ q=z_1$ has the form $(z_1,w)$ with $w\in P'$, and (6.8.7) for the holomorphic function $H(z_1,\cdot )$ on the ball $B(w,\delta )\subset P$ gives $|H(z_1,w)|^2\leq C_\delta \int _ P|H(z_1,w')|^2\, \mathrm dV(w')$, where $C_\delta ^{-1}$ is the volume of a ball of radius $\delta $ in $\mathbb C^{n-1}$. Summing over the $q$ points, we obtain $\sum _{t^ q=z_1}|H(\gamma (t))|^2\leq qC_\delta \int _ P|H(z_1,w')|^2\, \mathrm dV(w')$.

Now we integrate over $|z_1|<\rho ^ q$. The map $t\mapsto t^ q$ maps each of the $q$ half-open sectors $\{ 0<|t|<\rho ,\ 2\pi k/q\leq \arg t<2\pi (k+1)/q\} $ bijectively onto the punctured disc $0<|z_1|<\rho ^ q$, and on the interior of the sector it is a diffeomorphism onto the complement of a ray, with real Jacobian $|qt^{q-1}|^2$. By the change of variables formula, applied on each sector, $\int _{|t|<\rho }\Phi (t)q^2|t|^{2q-2}\, \mathrm dV_2(t)=\int _{|z_1|<\rho ^ q}\sum _{t^ q=z_1}\Phi (t)\, \mathrm dV_2(z_1)$ for every nonnegative measurable $\Phi $. Since $\Delta _{\rho ^ q}\times P\subset \Omega $, it follows that

with $c_0=1/(qC_\delta )$. If $n=1$, the same inequality holds with $c_0=1/q$, without the submean step.

Write $F\circ \gamma (t)=t^ mv(t)$ with an integer $m\geq 0$ and $v$ without zeros on $|t|\leq \rho $, and put $h=(H/F)\circ \gamma $. This function is holomorphic for $0<|t|<\rho $, because $F\circ \gamma $ has no zeros there, and by hypothesis it extends holomorphically across $0$ with $h(0)=0$; thus $h(t)=\sum _{\ell \geq 1}h_\ell t^\ell $ for $|t|<\rho $. Then $H\circ \gamma =t^ mvh$, and $|v|$ is bounded below by a positive constant on the compact disc $|t|\leq \rho $, being continuous and without zeros there. On each circle $|t|=s<\rho $ the power series of $h$ converges uniformly, and the monomials $t^\ell $ are orthogonal there, so $\int _0^{2\pi }|h(s\mathrm e^{\mathrm i\theta })|^2\, \mathrm d\theta =2\pi \sum _\ell |h_\ell |^2s^{2\ell }$. Integrating in polar coordinates and using the monotone convergence theorem, we obtain

with $c_1=\pi q^2c_0\min _{|t|\leq \rho }|v(t)|^2$. Now let $0<|a|<\rho ^ q$. For each of the $q$ roots $b$ of $b^ q=a$, the point $\gamma (b)$ lies on the slice $\{ z_1=a\} $, where $H=F$, and $F(\gamma (b))\neq 0$; hence $h(b)=1$. The average $q^{-1}\sum _{b^ q=a}b^\ell $ equals $a^{\ell /q}$ if $q$ divides $\ell $ and $0$ otherwise, so averaging $h$ over these roots gives $\sum _{j\geq 1}h_{qj}a^ j=1$. By the Cauchy–Schwarz inequality,

If $|a|\leq \rho ^ q/2$, the last sum equals $\rho ^{-2(m+q)}\sum _{j\geq 1}(qj+m+q)x^ j$ with $x=|a|^2\rho ^{-2q}\leq 1/4$, and this series is at most a constant times $x$. Hence (6.11.3) holds for $0<|a|<r=\rho ^ q/2$, with $c$ independent of $a$ and $H$. $\square$

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 $\mathcal I(\varphi )_ x$. $\square$

Proof. The assertions are local. On a relatively compact neighborhood of $x$ the weights are bounded above, and subtracting constants we may assume $\varphi \leq 0$ and $\chi \leq 0$ there; this changes $(1+\varepsilon )\varphi $ and $\varphi +\varepsilon \chi $ only by bounded amounts and hence changes none of the ideals. Then $(1+\varepsilon )\varphi $ and $\varphi +\varepsilon \chi $ decrease as $\varepsilon $ increases, so the corresponding ideals decrease with $\varepsilon $ and are contained in $\mathcal I(\varphi )_ x$. The sequences $(1+1/j)\varphi $ and $\varphi +\chi /j$ increase with $j$ and converge to $\varphi $ wherever $\varphi $ and $\chi $ are finite, hence almost everywhere. By theorem 6.11.4, $\mathcal I((1+1/j)\varphi )_ x=\mathcal I(\varphi )_ x$ and $\mathcal I(\varphi +\chi /j)_ x=\mathcal I(\varphi )_ x$ for all large $j$, which proves both assertions. $\square$