## Riemann Surfaces WS 2014/2015

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

A preliminary and probably very raw current version of the lecture notes: ps-file , pdf-file .
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(Sketches of) solutions of some exercises from the lecture course can be found here .
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 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.

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### 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.