Extending holomorphic motions and monodromy

TL;DR

This paper investigates when holomorphic motions of sets in the Riemann sphere over complex manifolds can be extended, establishing that trivial monodromy is both necessary and sufficient for such extensions over non-simply connected surfaces.

Contribution

It provides a topological criterion, trivial monodromy, for extending holomorphic motions over complex Riemann surfaces beyond the disk case.

Findings

Trivial monodromy is necessary and sufficient for extension.

Extension criteria depend on topological and geometric conditions.

Applications to lifting problems in Teichmüller theory.

Abstract

Let $E$ be a closed set in the Riemann sphere $\widehat{\mathbb{C}}$. We consider a holomorphic motion $\phi$ of $E$ over a complex manifold $M$, that is, a holomorphic family of injections on $E$ parametrized by $M$. It is known that if $M$ is the unit disk $\Delta$ in the complex plane, then any holomorphic motion of $E$ over $\Delta$ can be extended to a holomorphic motion of the Riemann sphere over $\Delta$. In this paper, we consider conditions under which a holomorphic motion of $E$ over a non-simply connected Riemann surface $X$ can be extended to a holomorphic motion of $\widehat{\mathbb{C}}$ over $X$. Our main result shows that a topological condition, the triviality of the monodromy, gives a necessary and sufficient condition for a holomorphic motion of $E$ over $X$ to be extended to a holomorphic motion of $\widehat{\mathbb{C}}$ over $X$. We give topological and geometric conditions for a holomorphic motion over a Riemann surface to be extended. We also apply our result to a lifting problem for holomorphic maps to Teichm\"uller spaces.