Riemann Surfaces WS 2012/2013

Wednesday, 11:30-13:00, room B27, Kampus Kirchberg


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.

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.

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.


Some relevant links


Topological classification of surfaces by Richard Koch.
A note on summability (in German) by Günther Trautmann.

Literature

Forster, Otto. - Lectures on Riemann surfaces; transl. by Bruce Gilligan, New York: Springer-Verlag, 1999.
Schlichenmaier, Martin. - An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces, Berlin: Springer, 2007.
Freitag, Eberhard; Busam, Rolf. - Complex analysis. Transl. from the German by Dan Fulea, Berlin: Springer, 2009.
Freitag, Eberhard. - Complex analysis 2. Riemann surfaces, several complex variables, Abelian functions, higher modular functions, Berlin: Springer, 2011.
Miranda, Rick. - Algebraic curves and Riemann surfaces, Providence RI: American Mathematical Society, 1995.
Farkas, Hershel M.; Kra, Irwin. - Riemann surfaces, Springer, 1992.
Griffiths, Philip; Harris, Joseph. - Principles of algebraic geometry, New York : J. Wiley, 1994.