Riemann Surfaces WS 2013/2014

Tuesday, 9:45-11:15, room B24a, Kampus Kirchberg
Formal description of the course: Description RS13-14.pdf


Some information on the exam

When: Friday, January 17, 2013


Where: room B04 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.

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 similar courses from WS 2011/2012 and WS 2012/2013 can be found here and here respectively.

Overview of the lectures and exercises

A preliminary and probably very raw current version of the lecture notes: ps-file , pdf-file .
Lectures: Exercises:
Lecture 1, Tuesday, September 17, 2013
Definition of Riemann surfaces, first examples and properties.
Exercises from Lecture 1 ps-file , pdf-file
Lecture 2, Tuesday, September 24, 2013
Meromorphic functions, first properties of morhisms of Riemann surfaces.
Exercises from Lecture 2 ps-file , pdf-file
Lecture 3, Tuesday, October 1, 2013
Elementary properties of morhisms of Riemann surfaces. Homotopy of curves, fundamental group.
Exercises from Lecture 3 ps-file , pdf-file
Lecture 4, Tuesday, October 8, 2013
Topological classification of compact Riemann surfaces, holomorphic maps between compact Riemann surfaces. Degree of a holomorphic map between compact Riemann surfaces.
Exercises from Lecture 4 ps-file , pdf-file
Lecture 5, Tuesday, October 15, 2013
Degree of a holomorphic map between compact Riemann surfaces. Divisors.
Exercises from Lecture 5 ps-file , pdf-file
Lecture 6, Tuesday, October 22, 2013
Sheaves of modules associated to divisors, Riemann-Roch space and its finite dimensionality.
Exercises from Lecture 6 ps-file , pdf-file
Lecture 7, Tuesday, October 29, 2013
Stalks of the structure sheaf, cotangent space, differentials.
Exercises from Lecture 7 ps-file , pdf-file
Lecture 8, Tuesday, November 5, 2013
Holomorphic and meromorphic differential forms.
Exercises from Lecture 8 ps-file , pdf-file
Lecture 9, Tuesday, November 12, 2013
Genus of a compact Riemann surface. The Riemann-Roch formula. The Riemann-Hurwitz fomula. Coverings and universal coverings. Holomorphic maps of complex tori.
Exercises from Lecture 9 ps-file , pdf-file
Lecture 10, Tuesday, November 19, 2013
Holomorphic maps of complex tori. Isomorphism classes of complex tori.
Exercises from Lecture 10 ps-file , pdf-file
Lecture 11, Tuesday, November 26, 2013
Meromorphic functions on complex tori.
Exercises from Lecture 11 ps-file , pdf-file
Lecture 12, Tuesday, December 3, 2013
Complex tori as algebraic curves. j-invariant.
Exercises from Lecture 12 ps-file , pdf-file
Lecture 13, Tuesday, December 10, 2013
Integration of differential forms along curves, residue theorem and its inverse.
Exercises from Lecture 13 ps-file , pdf-file
Lecture 14, Tuesday, December 17, 2013
Jacobian of a compact Riemann surface of positive genus. Abel-Jacobi theorem and its first corollaries. Compact Riemann surfaces as projective algebraic varieties.
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


Fundamental group. A chapter of the Algebraic topology book by Allen Hatcher.
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.