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https://hdl.handle.net/2440/103762
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Type: | Theses |
Title: | Holomorphic flexibility properties of spaces of elliptic functions |
Author: | Bowman, David James |
Issue Date: | 2016 |
School/Discipline: | School of Mathematical Sciences |
Abstract: | Let X be an elliptic curve and P the Riemann sphere. Since X is compact, it is a deep theorem of Douady that the set O(X, P) consisting of holomorphic maps X โ P admits a complex structure. If R๐ denotes the set of maps of degree n, then Namba has shown for n โฅ 2 that R๐ is a 2n-dimensional complex manifold. We study holomorphic flexibility properties of the spaces Rโ and Rโ. Firstly, we show that Rโ is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a 6-sheeted branched covering space of Rโ that is an Oka manifold. It follows that Rโ is C-connected and dominable. We show that Rโ is Oka if and only if Pโ\C is Oka, where C is a cubic curve that is the image of a certain embedding of X into Pโ. We investigate the strong dominability of Rโ and show that if X is not biholomorphic to C/ฮโ, where ฮโ is the hexagonal lattice, then Rโ is strongly dominable. As a Lie group, X acts freely on Rโ by precomposition by translations. We show that Rโ is holomorphically convex and that the quotient space Rโ/X is a Stein manifold. We construct an alternative 6-sheeted Oka branched covering space of Rโ 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. |
Advisor: | Larusson, Finnur Buchdahl, Nicholas |
Dissertation Note: | Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2016. |
Keywords: | oka manifolds mapping spaces elliptic functions several complex variables complex manifolds |
Provenance: | This electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at: http://www.adelaide.edu.au/legals |
DOI: | 10.4225/55/58c20ae708154 |
Appears in Collections: | Research Theses |
Files in This Item:
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01front.pdf | 319.64 kB | Adobe PDF | View/Open | |
02whole.pdf | 1.07 MB | Adobe PDF | View/Open | |
Permissions Restricted Access | Library staff access only | 229.68 kB | Adobe PDF | View/Open |
Restricted Restricted Access | Library staff access only | 1.1 MB | Adobe PDF | View/Open |
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