Holomorphic Flexibility Properties of Spaces of Elliptic Functions

TL;DR

This paper investigates the holomorphic flexibility of spaces of elliptic functions, showing that certain spaces are Oka manifolds, and explores their complex geometric properties including homogeneity, dominability, and convexity.

Contribution

It demonstrates that the space of degree 2 maps is homogeneous and Oka, and establishes that a branched cover of the degree 3 space is also Oka, advancing understanding of their complex geometric structures.

Findings

R_2 is a homogeneous Oka manifold.

A 6-sheeted branched cover of R_3 is Oka.

R_3 is C-connected and dominable.

Abstract

Let $X$ be an elliptic curve and $\mathbb{P}$ the Riemann sphere. Since $X$ is compact, it is a deep theorem of Douady that the set $\mathcal{O}(X,\mathbb{P})$ consisting of holomorphic maps $X\to \mathbb{P}$ admits a complex structure. If $R_n$ denotes the set of maps of degree $n$, then Namba has shown for $n\geq2$ that $R_n$ is a $2n$-dimensional complex manifold. We study holomorphic flexibility properties of the spaces $R_2$ and $R_3$. Firstly, we show that $R_2$ is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a $6$-sheeted branched covering space of $R_3$ that is an Oka manifold. It follows that $R_3$ is $\mathbb{C}$-connected and dominable. We show that $R_3$ is Oka if and only if $\mathbb{P}_2\backslash C$ is Oka, where $C$ is a cubic curve that is the image of a certain embedding of $X$ into $\mathbb{P}_2$. We investigate the strong dominability of $R_3$ and show that if $X$ is not biholomorphic to $\mathbb{C}/\Gamma_0$, where $\Gamma_0$ is the hexagonal lattice, then $R_3$ is strongly dominable. As a Lie group, $X$ acts freely on $R_3$ by precomposition by translations. We show that $R_3$ is holomorphically convex and that the quotient space $R_3/X$ is a Stein manifold. We construct an alternative $6$-sheeted Oka branched covering space of $R_3$ and prove that it is isomorphic to our first construction in a natural way. This alternative construction gives us an easier way of interpreting the fibres of the branched covering map.