## Riemann Surfaces WS 2015/2016

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

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