44BF9B7D0141
Vanishing to a given order at a point couples the variables through the total degree, even when the metric weight splits as a sum. On a compact product of curves, this coupling becomes a half-space condition on independent monomial indices, and the transition on each curve factor determines the local law in higher dimensions. An isolated pole on a submersion with connected fibers gives a further application: its multiplier ideal does not descend to the base, although the condition it imposes on pulled-back sections does.
Let $q\geq 2$. For $1\leq j\leq q$, choose integers $r_ j\geq 2$, at least two of them unequal, and numbers $\varepsilon _ j>0$. Put
\[ Y=(\mathbb P^1)^ q,\qquad x=(0,\ldots ,0),\qquad d_ j=r_ j+1,\qquad L=\boxtimes _ j\mathcal O_{\mathbb P^1}(d_ j). \]
In the affine coordinates $z_ j$ and the product frame $e$, put $t_ j=|z_ j|^2$ and define the smooth metric and volume by
\[ \begin{gathered} |e|_ h^2=\mathrm e^{-\phi },\qquad \phi =\sum _ j\psi _ j(t_ j),\qquad \psi _ j(t)=\log (1+t^{r_ j}+\varepsilon _ jt^{r_ j+1}),\\ \nu _ Y=a(t)\prod _ j\mathrm dA_ j,\qquad a(t)=\prod _ j a_ j(t_ j),\qquad a_ j(t)=(1+t)^{-2}, \end{gathered} \]
where $\mathrm dA_ j$ is Euclidean area in $z_ j$. In particular $\nu _ Y(Y)=\pi ^ q$ and $a(0)=1$. Define
\[ H_ j(t)=t\psi _ j'(t),\qquad G_ j(t)=tH_ j'(t),\qquad g_ j(t)=G_ j(t)/t,\qquad \theta =c_1(L,h). \]
The metric extends across infinity: there its weight, in terms of $u=|z_ j|^{-2}$, is $\log (\varepsilon _ j+u+u^{d_ j})$, which has positive curvature at infinity. For $t>0$, $H_ j(t)$ and $G_ j(t)$ are the mean and the variance of the exponents $0,r_ j,d_ j$ taken with probabilities proportional to $1,t^{r_ j},\varepsilon _ jt^{d_ j}$. Hence $G_ j>0$, and $H_ j$ increases from $0$ to $d_ j$. Near zero,
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18.44.1
\begin{equation} \begin{aligned} H_ j(t)& =r_ jt^{r_ j}+(r_ j+1)\varepsilon _ jt^{r_ j+1}+O(t^{2r_ j}),\\ G_ j(t)& =r_ j^2t^{r_ j}+(r_ j+1)^2\varepsilon _ jt^{r_ j+1}+O(t^{2r_ j}). \end{aligned} \end{equation}
Thus $\theta $ is semipositive, and its rank drops along the coordinate divisors $\{ z_ j=0\} $.
For integers $k_ m\to \infty $ with $k_ m=o(m)$, let $B_ m$ be the density of
\[ V_ m=H^0(Y,L^ m\otimes \mathfrak m_ x^{k_ m}),\qquad \| s\| ^2=\int _ Y|s|_{h^ m}^2\nu _ Y. \]
The ideal $\mathfrak m_ x^{k_ m}$ of the point $x$ imposes vanishing of total order $k_ m$, rather than separate orders in the individual variables. As before, $\Phi $ is the standard Gaussian distribution function.
Theorem 18.44.2.7840E247F9D5 Let $z=z^{(m)}$ be points of the affine chart, with $t_ j=|z_ j|^2$, such that $t_ j\to 0$ and $\mu _ j:=mH_ j(t_ j)$ satisfies $\mu _ j/k_ m\to \alpha _ j>0$, where $\sum _ j\alpha _ j=1$. Set $V=\sum _ jmG_ j(t_ j)$. If $(\sum _ j\mu _ j-k_ m)/\sqrt V\to \tau \in \mathbb R$, then
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18.44.3
\begin{equation} \frac{\pi ^ q a(t)B_ m(z)}{m^ q\prod _ jg_ j(t_ j)}\longrightarrow \Phi (\tau ). \end{equation}
The arguments of the coordinates $z_ j$ are arbitrary. For all sufficiently large $m$,
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18.44.4
\begin{equation} N_ m:=\dim V_ m=\prod _ j(d_ jm+1)-\binom {k_ m+q-1}{q}, \qquad N_ m/m^ q\longrightarrow D:=\prod _ jd_ j. \end{equation}
Moreover $m^{-q}B_ m\nu _ Y\to \theta ^ q/q!$ in total variation.
Proof.
The proof reduces the kernel to a probability for a sum of independent variables, one for each factor, and applies the one-variable transition of section 18.42 to each of them. In each variable, the affine coefficient of a global section of $L^ m$ is entire, and its expression in the frame at infinity must be holomorphic, so it is a polynomial of degree at most $d_ jm$. Hence the monomials $z^\ell e^ m$, $0\leq \ell _ j\leq d_ jm$, form a basis of the global sections. A polynomial vanishes to order at least $k_ m$ at $x$ exactly when all its monomials have total degree at least $k_ m$, so the monomials with $\sum _ j\ell _ j\geq k_ m$ form a basis of $V_ m$. Once $k_ m\leq \min _ j(d_ jm)+1$, which holds for large $m$, all monomials of total degree below $k_ m$ are global, and counting them gives (18.44.4).
Invariance under the rotation of each coordinate makes these monomials orthogonal. Their squared norms are $\prod _ jJ_{j,m,\ell _ j}$, where
\[ J_{j,m,\ell }=\pi \int _0^\infty t^\ell \mathrm e^{-m\psi _ j(t)}a_ j(t)\, \mathrm dt, \qquad w_{j,m,\ell }(t)=\frac{t^\ell \mathrm e^{-m\psi _ j(t)}}{J_{j,m,\ell }}. \]
These integrals converge even for $\ell =d_ jm$. Write $K_{j,m}=\sum _{\ell =0}^{d_ jm}w_{j,m,\ell }$, and let $\mathsf L_1,\ldots ,\mathsf L_ q$ be independent random variables with $\Pr \{ \mathsf L_ j=\ell \} =w_{j,m,\ell }(t_ j)/K_{j,m}(t_ j)$ for $0\leq \ell \leq d_ jm$. Then
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18.44.5
\begin{equation} B_ m(z)=\prod _ jK_{j,m}(t_ j)\, \Pr \left\{ \sum _ j\mathsf L_ j\geq k_ m\right\} . \end{equation}
Each factor $(\mathbb P^1,\mathcal O(d_ j))$, with the metric $\mathrm e^{-\psi _ j(|z_ j|^2)}$ and the volume $a_ j\, \mathrm dA_ j$, satisfies the hypotheses of theorem 18.42.15, and $K_{j,m}$ is its full density. Put $V_ j=mG_ j(t_ j)$. Since $\partial _{z_ j}\partial _{\bar z_ j}\psi _ j(|z_ j|^2)=g_ j(t_ j)$, the full density (18.42.25) gives $K_{j,m}=mg_ j(t_ j)/(\pi a_ j(t_ j))(1+o(1))$. For any fixed $y\in \mathbb R$, apply the radial formula (18.42.33) to the integer order
\[ b_{j,m}=\lfloor \mu _ j+y\sqrt{V_ j}\rfloor +1. \]
This order tends to infinity and is sublinear. The selected kernel of the factor, divided by its full kernel, is $\Pr \{ \mathsf L_ j\geq b_{j,m}\} $, and it tends to $\Phi (-y)$. Hence
\[ \frac{\mathsf L_ j-\mu _ j}{\sqrt{V_ j}}\ \Longrightarrow \ N(0,1). \]
That is, these normalized variables converge in distribution to a standard Gaussian variable. Here $\mu _ j$ and $\sqrt{V_ j}$ serve only as centering and scale; we do not need the exact mean or variance of $\mathsf L_ j$. Since $V_ j/V\to r_ j\alpha _ j/\sum _ i r_ i\alpha _ i>0$, independence gives a standard Gaussian limit for $(\sum _ j\mathsf L_ j-\sum _ j\mu _ j)/\sqrt V$. To see this, use the product of the limiting Gaussian laws on bounded rectangles, and then tightness and the fact that the boundary of a half-space has zero Gaussian measure. Consequently the probability in (18.44.5) tends to $\Phi (\tau )$, which proves the local assertion for every sequence $k_ m\to \infty $ with $k_ m=o(m)$.
For the global assertion we also need the full kernel at a fixed $t>0$, where the formula for shrinking radii does not apply. We suppress the index of the factor and set $F(u)=\psi (\mathrm e^ u)$. Then $F'(u)=H(\mathrm e^ u)$ and $F''(u)=G(\mathrm e^ u)>0$. For $\ell /m$ in a compact subinterval of $(0,d)$, the saddle $s=H^{-1}(\ell /m)$ stays compact, and Laplace expansion gives uniformly
\[ J_{m,\ell }=\pi a(s)s\sqrt{\frac{2\pi }{mG(s)}} \mathrm e^{\ell \log s-m\psi (s)}(1+o(1)). \]
Indeed, the phase in the logarithmic variable has a uniformly positive second derivative there, and outside a fixed neighborhood of the saddle point, convexity and the slopes at the endpoints bound the tails exponentially. For $\ell =mH(t)+O(\sqrt m)$, evaluation yields
\[ w_{m,\ell }(t)=\frac{\sqrt{mG(t)}}{\pi a(t)t\sqrt{2\pi }} \mathrm e^{-(\ell -mH(t))^2/(2mG(t))}(1+o(1)). \]
For indices with saddle points in a compact interval, the loss in the phase, as a function of $\ell $, has second derivative $1/(mG(s))\geq c/m$, which gives summable Gaussian tails. For the remaining indices, fix small numbers $\delta ,\sigma >0$, and bound the norm $J_{m,\ell }$ from below by the integral over an interval of length $1/m$ about $t\mathrm e^{-\sigma }$ or $t\mathrm e^{\sigma }$. If $|\ell /m-H(t)|\geq \delta $, the appropriate choice gains at least $m(\delta \sigma -C\sigma ^2)\geq cm$ in the exponent. Each such weight $w_{m,\ell }(t)$ is then at most $Cm\mathrm e^{-cm}$, and there are $dm+1$ indices. Gaussian summation therefore gives $K_ m(t)/m\to g(t)/(\pi a(t))$.
In the product chart, $\theta ^ q/q!=\prod _ j(g_ j(t_ j)/\pi )\mathrm dA_ j$, and its mass is $D$ because $\int _0^\infty g_ j(t)\, \mathrm dt=H_ j(\infty )-H_ j(0)=d_ j$. The normalized full density $m^{-q}\prod _ jK_{j,m}$ therefore tends to the density of $\theta ^ q/q!$ with respect to $\nu _ Y$ almost everywhere; the coordinate divisors have zero volume. The total mass of the normalized full kernel measure is $m^{-q}\prod _ j(d_ jm+1)\to D$. For the nonnegative densities $f_ m,f$ of these measures with respect to $\nu _ Y$, dominated convergence applied to $\min (f_ m,f)$ gives
\[ \int |f_ m-f|\nu _ Y =\int f_ m\nu _ Y+\int f\nu _ Y-2\int \min (f_ m,f)\nu _ Y\longrightarrow 0. \]
Finally, the difference between the full and the selected kernel is nonnegative and has integral $\binom {k_ m+q-1}{q}=o(m^ q)$. This proves the total-variation convergence.
$\square$
Pome
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