Postojanje meromorfnih funkcija na Riemannovim plohama

Milivojević, Aleksandar (2015) Postojanje meromorfnih funkcija na Riemannovim plohama. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

This thesis is an introduction to the theory of Riemann surfaces. We define Riemann surfaces and holomorphic maps between them, and prove some basic properties. We then apply complex analytic techniques to address the question of existence of meromorphic functions on an arbitrary Riemann surface. We study differential 1-forms and decompose the Hilbert space of square-integrable 1-forms. Using this decomposition, we prove that there exist meromorphic 1-forms with prescribed singularities, whence we obtain the existence of meromorphic functions as well. Our focus then narrows to compact Riemann surfaces. We define algebraic curves, and then briefly expound the necessary theory to state the Riemann-Roch theorem, which links the degree of a divisor to the dimension of a related meromorphic function space on an algebraic curve. Applying this theorem, we prove that every algebraic curve can be embedded into projective space. We then comment on the fact that the previously proven result on the existence of meromorphic functions on Riemann surfaces implies that all Riemann surfaces are algebraic curves.

Item Type: Thesis (Diploma thesis)
Supervisor: Muić, Goran
Date: 2015
Number of Pages: 81
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 23 Oct 2015 08:49
Last Modified: 23 Oct 2015 08:49
URI: http://digre.pmf.unizg.hr/id/eprint/4173

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