Riemann Surfaces WS 2014/2015

Tuesday, 9:45-11:15, room B24a, Kampus Kirchberg
Formal description of the course: Description RS14-15.pdf


Some information on the exam

When: Friday, January 16, 2015


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.
There was an oral exam at the end of the course.
Some information on similar courses from WS 2011/2012, WS 2012/2013, and WS 2013/2014 can be found here, 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 .

(Sketches of) solutions of some exercises from the lecture course can be found here .

Lectures: Exercises:
Lecture 1, Tuesday, September 16, 2014
Definition of Riemann surfaces, first examples. Holomorhic functions.
Exercises from Lecture 1 ps-file , pdf-file
Lecture 2, Tuesday, September 23, 2014
Basic properies of holomorphic functions. Meromorphic functions, first properties of morhisms of Riemann surfaces.
Exercises from Lecture 2 ps-file , pdf-file
Lecture 3, Tuesday, September 30, 2014
Elementary properties of morphisms of Riemann surfaces.
Exercises from Lecture 3 ps-file , pdf-file
Lecture 4, Tuesday, October 7, 2014
Fundamental group, topological classification of compact Riemann surfaces.
Exercises from Lecture 4 ps-file , pdf-file
Lecture 5, Tuesday, October 14, 2014
Degree of a holomorphic map between compact Riemann surfaces. Number of poles and zeroes of meromorphic functions on compact Riemann surfaces.
Exercises from Lecture 5 ps-file , pdf-file
Lecture 6, Tuesday, October 21, 2014
Divisors. Sheaves of modules associated to divisors, Riemann-Roch space.
Exercises from Lecture 6 ps-file , pdf-file
Lecture 7, Tuesday, October 28, 2014
Finite dimensionality of the Riemann-Roch spaces on compact Riemann surfaces. Stalks of the structure sheaf.
Exercises from Lecture 7 ps-file , pdf-file
Lecture 8, Tuesday, November 4, 2014
Cotangent space, differentials. Holomorphic and meromorphic differential forms and their divisors.
Exercises from Lecture 8 ps-file , pdf-file
Lecture 9, Tuesday, November 11, 2014
Relation between global meromorphic functions and differential forms. Genus of a compact Riemann surface. The Riemann-Roch formula. The Riemann-Hurwitz fomula.
Exercises from Lecture 9 ps-file , pdf-file
Lecture 10, Tuesday, November 18, 2014
Coverings and universal coverings. Holomorphic maps of complex tori. Isomorphism classes of complex tori.
Exercises from Lecture 10 ps-file , pdf-file
Lecture 11, Tuesday, November 25, 2014
Automorphisms of complex tori. Meromorphic functions on complex tori.
Exercises from Lecture 11 ps-file , pdf-file
Lecture 12, Tuesday, December 2, 2014
Complex tori as algebraic curves. j-invariant.
Exercises from Lecture 12 ps-file , pdf-file
Lecture 13, Tuesday, December 9, 2014
Integration of differential forms along curves, residue theorem and its inverse.
Exercises from Lecture 13 ps-file , pdf-file
Lecture 14, Tuesday, December 16, 2014
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.