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On the dimension of Bergman spaces on $\mathbb{P}^1$

Anne-Katrin Gallagher, Purvi Gupta, Liz Vivas

Abstract

Inspired by a result by Szőke, we give potential-theoretic characterizations of the dimension of the Bergman space of holomorphic sections of a restriction of a holomorphic line bundle of $\mathbb{P}^1$ to some open set $D\subset\mathbb{P}^1$.

On the dimension of Bergman spaces on $\mathbb{P}^1$

Abstract

Inspired by a result by Szőke, we give potential-theoretic characterizations of the dimension of the Bergman space of holomorphic sections of a restriction of a holomorphic line bundle of to some open set .

Paper Structure

This paper contains 13 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Bergman spaces in $\mathbb{C}^n$ and potential theory in $\mathbb{C}$
  4. Bergman spaces of holomorphic sections
  5. Line bundles on $\mathbb{P}^1$
  6. Potential theory on $\mathbb{P}^1$
  7. Proof(s) of the main result
  8. Proof of $\mathbf{(a)}$ implies $\mathbf{(b)}$
  9. Proof of $\mathbf{(c)}$ implies $\mathbf{(a)}$ via the Cauchy transform approach
  10. Proof of $\mathbf{(c)}$ implies $\mathbf{(d)}$
  11. Proof of $\mathbf{(d)}$ implies $\mathbf{(a)}$
  12. A corollary to Lemma \ref{['L:GaHaHemodification']} on weighted Bergman spaces
  13. Finite dimensional Bergman spaces in $\mathbb P^2$

Key Result

Theorem 1.1

Let $K\subsetneq\mathbb{C}$ be a closed subset. Then the following are equivalent.

Theorems & Definitions (24)

  • Theorem 1.1: Ca67,Wi84,GaHaHe17,GaLeRa21
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 14 more