Postojanje meromorfnih funkcija na Riemannovim plohama

master's thesis

• Author: Aleksandar Milivojević
• Mentor: Goran Muić (mentor)
• Committee member: Goran Muić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Željka Milin-Šipuš (član povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-10, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad je uvod u teoriju Riemannovih ploha. Definiramo Riemannove plohe i holomorfna preslikavanja medu njima i dokazujemo neka njihova osnovna svojstva. Zatim tehnikama iz kompleksne analize pristupamo pitanju postojanja meromorfnih funkcija na proizvoljnoj Riemannovoj plohi. Prouˇcavamo diferencijalne 1-forme i činimo dekompoziciju Hilbertovog prostora kvadratno integrabilnih 1-formi. Pomoću te dekompozicije dokazujemo da postoje meromorfne 1-forme s unaprijed zadanim singularitetima, odakle dobivamo i meromorfne funkcije. Zatim se usredotočimo na kompaktne Riemannove plohe. Definiramo algebarske krivulje, te ukratko izlažemo teoriju potrebnu da bi se iskazao Riemann-Rochov teorem, koji povezuje stupanj divizora i dimenziju pripadnog prostora meromorfnih funkcija na algebarskoj krivulji. Pomoću tog teorema dokazujemo da se svaka algebarska krivulja može uložiti u projektivan prostor, te komentiramo da prethodno dokazan rezultat o postojanju meromorfnih funkcija na Riemannovoj plohi povlači da je svaka kompaktna Riemannova ploha algebarska krivulja.

Abstract (english)

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.

• Keywords: meromorfne funkcije; Riemannove plohe; holomorfna preslikavanja; Riemann-Rochov teorem
• Keywords (english): meromorphic functions; Riemann surfaces; holomorphic maps; Riemann-Roch theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:680399
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-24 12:48:46