Riemann Surfaces WS 2015/2016

Tuesday, 11:30-13:00, room B23, Kampus Kirchberg
Formal description of the course: Description RS14-15.pdf


Information concerning the (repetition) exam in June 2016.

When: Monday, June 13, 2016, from 9:00 till 10:00


Where: B 02 CK 

Some information on the exam

When: Monday, January 18, 2016


Where: room B04 CK







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, WS 2012/2013, WS 2013/2014, and WS 2014/2015 can be found here, 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 .

Lectures: Exercises:
Lecture 1, Tuesday, September 15, 2016
Definition of Riemann surfaces. First examples.
This lecture was given by Dr. Sara Arias-de-Reyna .
Exercises from Lecture 1 ps-file , pdf-file
Lecture 2, Tuesday, September 22, 2015
Holomorphic functions, holomorphic maps, meromorphic functions, first properties of morhisms of Riemann surfaces.
Exercises from Lecture 2 ps-file , pdf-file
Lecture 3, Tuesday, September 29, 2015
Meromorphic functions as holomorphic maps to the Riemann sphere, local behaviour of holomorphic maps.
Exercises from Lecture 3 ps-file , pdf-file
Lecture 4, Tuesday, October 6, 2015
Local behaviour of holomorphic maps: first corollaries. Fundamental group, topological classification of compact Riemann surfaces.
Exercises from Lecture 4 ps-file , pdf-file
Lecture 5, Tuesday, October 13, 2015
Topological classification of compact Riemann surfaces. 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 20, 2015
Divisors. Sheaves of modules associated to divisors.
Exercises from Lecture 6 ps-file , pdf-file
Lecture 7, Tuesday, October 27, 2015
Divisors and invertible sheaves. Riemann-Roch space and its finite dimensionality.
Exercises from Lecture 7 ps-file , pdf-file
Lecture 8, Tuesday, November 3, 2015
Cotangent space, differentials. Holomorphic differential forms.
Exercises from Lecture 8 ps-file , pdf-file
Lecture 9, Tuesday, November 10, 2015
Meromorphic differential forms. 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 17, 2015
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 24, 2015
Automorphisms of complex tori. Meromorphic functions on complex tori.
Exercises from Lecture 11 ps-file , pdf-file
Lecture 12, Tuesday, December 1, 2015
Complex tori as algebraic curves. j-invariant.
Exercises from Lecture 12 ps-file , pdf-file
Lecture 13, Tuesday, December 8, 2015
Integration of differential forms along curves, residue theorem and its inverse.
Exercises from Lecture 13 ps-file , pdf-file
Lecture 14, Tuesday, December 15, 2015
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.