Fix an integer $r\geq 1$ and, for each positive integer $m$, a nonnegative integer $q_ m$. The model of section 18.41 on $\mathbb C$ has the smooth weight $|z|^{2r}$ and the selecting ideals $(z^{q_ m})$. When $q_ m\to \infty $, the kernel vanishes in the natural rescaled limit, but it has a Gaussian transition at the larger radius $(q_ m/(rm))^{1/(2r)}$. For $q_ m=\lfloor m\lambda \rfloor $ with a real $\lambda >0$, the limiting kernel measure misses the atom at the pole of the limiting envelope current. On a compact curve with the curvature and local expansion of theorem 18.42.1, that theorem transfers the Gaussian transition to global sections when $q_ m=o(m^{1/(2r+1)})$, and the construction with curvature boundaries in theorem 18.42.15 extends the corrected transition to every sublinear order. We record the original question; it is answered by theorem 18.42.27 together with the corrections from the weight in subsection 18.42.30.
Problem 24.10.4.4C662988834F Let $C$ be a connected compact Riemann surface, let $x\in C$, and let $(L,h)$ be a smooth Hermitian holomorphic line bundle whose curvature is semipositive and strictly positive on $C\setminus \{ x\} $. Fix an integer $r\geq 1$. Suppose that in a local coordinate $z$ centered at $x$ and a holomorphic frame $e$ one has
\[ |e|_ h^2=\mathrm{e}^{-\phi },\qquad \phi (z)=|z|^{2r}+O(|z|^{2r+1}). \]
Let $\nu $ be a smooth positive volume form with local expression $a(z)\, \mathrm{d}A$ and $a(0)=1$, where $\mathrm{d}A$ is Euclidean area. For positive integers $m$, choose integers $q_ m\to \infty $ with $q_ m=o(m)$. Give
\[ V_ m=H^0(C,L^ m\otimes \mathfrak m_ x^{q_ m}) \]
the norm $\| s\| ^2=\int _ C|s|_{h^ m}^2\, \mathrm{d}\nu $, where $\mathfrak m_ x$ is the ideal sheaf of $x$. Write $B_ m=\sum _ j|s_{m,j}|_{h^ m}^2$ for an orthonormal basis $(s_{m,j})$ of $V_ m$, with value zero when $V_ m=0$. Determine the growth ranges of $q_ m$ for which, for each fixed real $\tau $, every sequence of points $x_ m$ in the coordinate neighborhood satisfying
\[ m|z(x_ m)|^{2r}=q_ m/r+\tau \sqrt{q_ m/r} \]
obeys
\[ \frac{\pi B_ m(x_ m)}{mr^2|z(x_ m)|^{2r-2}} \longrightarrow \Phi (\tau ):=\frac1{\sqrt{2\pi }}\int _{-\infty }^{\tau } \mathrm{e}^{-s^2/2}\, \mathrm{d}s. \]
When this transition law fails, determine how further terms of the local weight change its center and scale.
For the data of this problem, theorem 18.42.27 shows that the law holds in the optimal universal range $q_ m=o(m^{1/(r+1)})$, uniformly in the direction of observation; outside that range, a compact example obtained by a holomorphic change of coordinate violates the uncorrected law along a subsequence. For a fixed weight, the exact criterion is (18.42.29).
In the whole range $q_ m\to \infty $, $q_ m=o(m)$, theorem 18.42.15 centers the transition on the boundary of the curvature quadrature domain and uses the normal length scale $1/(2\sqrt{m\rho })$, where $\rho $ is the density of the curvature. The first effective Taylor coefficient and the ranges of the corrections of finite order are computed in subsection 18.42.30; a flat perturbation there shows that no fixed finite Taylor jet of the weight determines the center to within the width of the transition for every sublinear sequence. This resolves the problem on compact curves as stated.
In higher dimension, for ordinary point jets of growing total order on the explicit anisotropic product of section 18.44, theorem 18.44.2 proves a Gaussian transition throughout the sublinear range of orders. The correct radial center includes the nonzero remainder terms of the weight, whereas proposition 18.44.8 gives the sharp, smaller range in which the homogeneous center can be used in this class. Theorem 18.44.11 transfers these laws through a submersion with compact connected fibers, with a pulled-back metric and a matched smooth volume; there the isolated selecting weight varies with the tensor power. Mixed or angular remainders, observation near the coordinate faces, other bases and general transfers through fiber selectors remain open directions.
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