Riemann Surfaces WS 2011/2012

Tuesday, 11:30-13:00, room B24b, Kampus Kirchberg


Information concerning the (repetition) exam in June 2012.

When: Wednesday, June 13, 2012, from 14:00 till 16:00


Where: B 27 CK 

Some information concerning the exam.

When: Wednesday, January 25, 2012


Where: B 13 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.


Exercises from the lecture course

ps-file
,

pdf-file





Lectures: Exercises:
Lecture 1, Tuesday, September 20, 2011
Definition of Riemann surfaces, first examples and properties.
Exercises from Lecture 1 ps-file , pdf-file
Lecture 2, Tuesday, September 27, 2011
Meromorphic functions, elementary properties of holomorphic maps.
Exercises from Lecture 2 ps-file , pdf-file
Lecture 3, Tuesday, October 4, 2011
Homotopy of curves, fundamental group, coverings, holomorphic maps of Riemann surfaces as ramified coverings.
Exercises from Lecture 3 ps-file , pdf-file
Lecture 4, Tuesday, October 11, 2011
Universal coverings, deck transformations, divisors.
Exercises from Lecture 4 ps-file , pdf-file
Lecture 5, Tuesday, October 18, 2011
Sheaves of modules associated to divisors, Riemann-Roch space and its finite dimensionality.
Exercises from Lecture 5 ps-file , pdf-file
Lecture 6, Tuesday, October 25, 2011
Holomorpic and meromorphic differential forms.
Exercises from Lecture 6 ps-file , pdf-file
Lecture 7, Tuesday, November 8, 2011
The theorem of Riemann-Roch, first corollaries, meromorphic functions on complex tori.
Exercises from Lecture 7 ps-file , pdf-file
Lecture 8, Tuesday, November 15, 2011
Field of meromorphic functions on complex tori, complex tori as plane projective curves.
Correction: description of even elliptic functions with the poles on the lattice: ps-file , pdf-file.
Exercises from Lecture 8 ps-file , pdf-file
Lecture 9, Tuesday, November 22, 2011
Integration of differential forms along curves, residue theorem.
Correction: mistake in the definition of integral: ps-file , pdf-file.
Exercises from Lecture 9 ps-file , pdf-file
Lecture 10, Tuesday, November 29, 2011
Corollaries of the residue theorem. Proof of the Riemann-Roch theorem: preparatory work.
Exercises from Lecture 10 ps-file , pdf-file
Lecture 11, Tuesday, December 6, 2011
Topological classification of compact Riemann surfacess, periods, proof of the Riemann-Roch theorem, Jacobian of a compact Riemann surface.
Exercises from Lecture 11 ps-file , pdf-file
Lecture 12, Tuesday, December 13, 2011
Abel's theorem and its first corollaries, Jacobi's theorem, remarks on the classification of compact Riemann surfaces of genus 1, divisors and invertible sheaves.
Exercises from Lecture 12 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.