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Projective embedding of stably degenerating sequence of hyperbolic Riemann surfaces

Jingzhou Sun

Abstract

Given a sequence of genus $g\geq 2$ curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature $-1$. we prove that the Kodaira embedding using orthonormal basis of the Bergman space of sections of a pluri-canonical bundle also converges to the embedding of the limit space together with extra complex projective lines.

Projective embedding of stably degenerating sequence of hyperbolic Riemann surfaces

Abstract

Given a sequence of genus curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature . we prove that the Kodaira embedding using orthonormal basis of the Bergman space of sections of a pluri-canonical bundle also converges to the embedding of the limit space together with extra complex projective lines.

Paper Structure

This paper contains 4 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Punctured Model
  3. the Collar model
  4. convergence of projective embedding

Key Result

Theorem 1.1

For $k$ large enough, we can choose an orthonormal basis for $\mathcal{H}_{j,k}$ for all $j>0$, so that as $j\to \infty$ the image of the embedding induced by the orthonormal basis converges to the image of $C_0$ under the embedding attached with $d$ pairs of linear ${\mathbb C}{\mathbb P}^1$'s. To each pair of the ends $(p_\alpha,p_{\alpha+d})$, a pair of linear ${\mathbb C}{\mathbb P}^1$'s are

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2