Algebraic Oka Theory for Curves
Date
2022
Authors
Dye, Ryan James
Editors
Advisors
Larusson, Finnur
Stevenson, Daniel
Stevenson, Daniel
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Thesis
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Abstract
In 1989, Winkelmann gave a complete classification of the pairs of Riemann surfaces (X, Y ) that satisfy the basic Oka principle (BOP): every continuous map from X into Y is homotopic to a holomorphic one. This thesis is motivated by the analogous problem in the algebraic category. Explicitly, we seek to provide a complete classification of the pairs of smooth algebraic curves that satisfy the basic algebraic Oka principle (aBOP). We give an elementary proof of a special case of Serre’s seminal GAGA principle, which will allow us to translate the problem thus formulated in algebro-geometric terms, into the complex-geometric domain. Here we can use Winkelmann’s result to assist us with our goal. In particular, a pair of smooth algebraic curves cannot satisfy aBOP if it does not satisfy BOP. Finally, we discuss a framework for which the basic algebraic Oka property with approximation can be explored.
School/Discipline
School of Mathematical Sciences
Dissertation Note
Thesis (MPhil) -- University of Adelaide, School of Mathematical Sciences, 2022
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