## Riemann Surfaces WS 2012/2013

Wednesday, 11:30-13:00, room B27, Kampus Kirchberg
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### Some general remarks on the course

There was one lecture every week. Every lecture was provided with a list of relevant exercises useful for understanding of the lecture.
Some information on a similar course from WS 2011/2012 can be found here.
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### Some information on the exam

When: Friday, January 18, 2013

Where: B 14 CK

A detailed list of covered topics :

ps-file
,

pdf-file
.

General remarks.

Every notion is supposed to be provided with an example.
Every statement should be also illustrated by an example of its application.
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### Overview of the lectures and exercises

Exercises from the lecture course: ps-file , pdf-file .
 Lectures: Exercises: Lecture 1, Wednesday, September 19, 2012 Definition of Riemann surfaces, first examples and properties. This lecture was presented by Dr. Simon Brain . Exercises from Lecture 1 ps-file , pdf-file Lecture 2, Wednesday, September 26, 2012 Meromorphic functions, elementary properties of morhisms (holomorphic maps) of Riemann surfaces. Exercises from Lecture 2 ps-file , pdf-file Lecture 3, Wednesday, October 3, 2012 Homotopy of curves, fundamental group, topological classification of compact Riemann surfaces, holomorphic maps between compact Riemann surfaces. Exercises from Lecture 3 ps-file , pdf-file Lecture 4, Wednesday, October 10, 2012 Degree of a holomorphic map between compact Riemann surfaces, divisors. Exercises from Lecture 4 ps-file , pdf-file Lecture 5, Wednesday, October 17, 2012 Sheaves of modules associated to divisors, Riemann-Roch space and its finite dimensionality. Exercises from Lecture 5 ps-file , pdf-file Lecture 6, Wednesday, October 24, 2012 Holomorphic differential forms. Exercises from Lecture 6 ps-file , pdf-file Lecture 7, Wednesday, October 31, 2012 Examlpe session. Meromorphic differential forms. Exercises from Lecture 7 ps-file , pdf-file Lecture 8, Wednesday, November 7, 2012 Genus of a compact Riemann surface. The Riemann-Roch formula, some examples. The Riemann-Hurwitz fomula. Exercises from Lecture 8 ps-file , pdf-file Lecture 9, Wednesday, November 14, 2012 Existence of non-constant meromorphic functions. Coverings and universal coverings. Holomorphic maps of complex tori. Exercises from Lecture 9 ps-file , pdf-file Lecture 10, Wednesday, November 21, 2012 The space of complex tori as an "almost" Riemann surface. Meromorphic functions on complex tori. Correction: there was a mistake in the lecture. Please read carefully the statement of Exercise 40 and compare it with the corresponding statement from the lecture. Exercises from Lecture 10 ps-file , pdf-file Lecture 11, Wednesday, November 28, 2012 Field of meromorphic functions on complex tori, even and odd elliptic functions, complex tori as plane projective curves. Exercises from Lecture 11 ps-file , pdf-file Lecture 12, Wednesday, December 5, 2012 j-invariant. Integration of differential forms along curves, residue theorem and its inverse. Exercises from Lecture 12 ps-file , pdf-file Lecture 13, Wednesday, December 12, 2012 Residues of differential forms and existence of non-constant meromorphic functions. Jacobian of a compact Riemann surface of positive genus. Exercises from Lecture 13 ps-file , pdf-file Lecture 14, Wednesday, December 19, 2012 Abel-Jacobi theorem and its first corollaries. Compact Riemann surfaces as projective algebraic varieties. Genus of plane smooth algebraic curves. Exercises from Lecture 14 ps-file , pdf-file

Please let me know about misprints and/or mistakes found in the exercises and the lectures.

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