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Publication . Preprint . Article . 2016 . Embargo end date: 01 Jan 2016

Holomorphic Flexibility Properties of Spaces of Elliptic Functions

David Bowman;
Open Access
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.
Subjects by Vocabulary

arXiv: Mathematics::Complex Variables

Microsoft Academic Graph classification: Branched covering Combinatorics Mathematics Holomorphic function Riemann sphere symbols.namesake symbols Image (category theory) Quotient space (linear algebra) Stein manifold Degree (graph theory) Complex manifold

Subjects

Complex Variables (math.CV), FOS: Mathematics, Mathematics - Complex Variables, General Mathematics

1 Introduction 1 1.1 Background and Context . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Results of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Further Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Proof of Main Theorem 11 2.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 The Universal Complex Structure . . . . . . . . . . . . . . . . 13 2.2.2 Symmetric Products . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.3 The Jacobi Map . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 The Theta Function . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.5 The Divisor Map . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.6 Nonsingularity of Rn . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.7 The Space Rn=M . . . . . . . . . . . . . . . . . . . . . . . . . 27 [13] Freitag, E. and Busam, R. Complex analysis. Springer, 2005.

[22] Kaup, L. and Kaup, B. Holomorphic functions of several variables. De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter, 1983. [OpenAIRE]

[23] Kaup, W. Holomorphic mappings of complex spaces. Symposia Mathematica 2 (1968) 333{340.

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