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A simple remark on holomorphic maps on Torelli space of marked spheres

Ruben A. Hidalgo

Abstract

The configuration space of $k \geq 3$ ordered points in the Riemann sphere $\widehat{\mathbb C}$ is the Torelli space ${\mathcal U}_{0,k}$; a complex manifold of dimension $k-3$. If $m,n \geq 4$ and $F:{\mathcal U}_{0,m} \to {\mathcal U}_{0,n}$ is a non-constant holomorphic map, then we observe that (i) $n \leq m$ and (ii) each coordinate of $F$ is given by a cross-ratio.

A simple remark on holomorphic maps on Torelli space of marked spheres

Abstract

The configuration space of ordered points in the Riemann sphere is the Torelli space ; a complex manifold of dimension . If and is a non-constant holomorphic map, then we observe that (i) and (ii) each coordinate of is given by a cross-ratio.

Paper Structure

This paper contains 13 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Torelli's space ${\mathcal{U}}_{0,k+1}$
  4. An explicit model of ${\mathcal{U}}_{0,k+1}$ and its automorphisms
  5. Forms of the coordinates $T_{j}^{\sigma}$ of $\Theta_{k}$
  6. Auxiliary facts
  7. Example
  8. The Picard Theorem for Riemann surfaces
  9. Proof of Theorem \ref{['main']}
  10. Case $m=n=3$
  11. Case $m=3$ and $n \geq 4$
  12. Case $n=3$ and $m \geq 4$
  13. Case $m,n \geq 4$

Key Result

Theorem 1

Let $F:\Omega_{m} \to \Omega_{n}$, where $n,m \geq 3$ are integers, a non-constant holomorphic map. Then

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 1 more