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

TL;DR

This paper studies holomorphic maps between Torelli spaces of marked spheres, showing such maps are constrained by dimension and are expressed via cross-ratios, revealing structural rigidity.

Contribution

It establishes that non-constant holomorphic maps between Torelli spaces are dimensionally restricted and are explicitly given by cross-ratios, highlighting their rigid structure.

Findings

If $F$ is non-constant, then $n \

each coordinate of $F$ is a cross-ratio

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.