## Riemann Surfaces WS 2011/2012

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

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pdf-file

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 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
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Please let me know about misprints and/or mistakes found in the exercises and the lectures.
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