Problem 24.10.44C662988834F

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

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

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

obeys

When this transition law fails, determine how further terms of the local weight change its center and scale.