Riemann Surfaces WS 2013/2014

Tuesday, 9:45-11:15, room B24a, Kampus Kirchberg
Formal description of the course: Description RS13-14.pdf
```

```

Some information on the exam

When: Friday, January 17, 2013

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.
Some information on similar courses from WS 2011/2012 and WS 2012/2013 can be found 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 17, 2013 Definition of Riemann surfaces, first examples and properties. Exercises from Lecture 1 ps-file , pdf-file Lecture 2, Tuesday, September 24, 2013 Meromorphic functions, first properties of morhisms of Riemann surfaces. Exercises from Lecture 2 ps-file , pdf-file Lecture 3, Tuesday, October 1, 2013 Elementary properties of morhisms of Riemann surfaces. Homotopy of curves, fundamental group. Exercises from Lecture 3 ps-file , pdf-file Lecture 4, Tuesday, October 8, 2013 Topological classification of compact Riemann surfaces, holomorphic maps between compact Riemann surfaces. Degree of a holomorphic map between compact Riemann surfaces. Exercises from Lecture 4 ps-file , pdf-file Lecture 5, Tuesday, October 15, 2013 Degree of a holomorphic map between compact Riemann surfaces. Divisors. Exercises from Lecture 5 ps-file , pdf-file Lecture 6, Tuesday, October 22, 2013 Sheaves of modules associated to divisors, Riemann-Roch space and its finite dimensionality. Exercises from Lecture 6 ps-file , pdf-file Lecture 7, Tuesday, October 29, 2013 Stalks of the structure sheaf, cotangent space, differentials. Exercises from Lecture 7 ps-file , pdf-file Lecture 8, Tuesday, November 5, 2013 Holomorphic and meromorphic differential forms. Exercises from Lecture 8 ps-file , pdf-file Lecture 9, Tuesday, November 12, 2013 Genus of a compact Riemann surface. The Riemann-Roch formula. The Riemann-Hurwitz fomula. Coverings and universal coverings. Holomorphic maps of complex tori. Exercises from Lecture 9 ps-file , pdf-file Lecture 10, Tuesday, November 19, 2013 Holomorphic maps of complex tori. Isomorphism classes of complex tori. Exercises from Lecture 10 ps-file , pdf-file Lecture 11, Tuesday, November 26, 2013 Meromorphic functions on complex tori. Exercises from Lecture 11 ps-file , pdf-file Lecture 12, Tuesday, December 3, 2013 Complex tori as algebraic curves. j-invariant. Exercises from Lecture 12 ps-file , pdf-file Lecture 13, Tuesday, December 10, 2013 Integration of differential forms along curves, residue theorem and its inverse. Exercises from Lecture 13 ps-file , pdf-file Lecture 14, Tuesday, December 17, 2013 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.

```
```