Department of Mathematics
Joachim Wehler
LMU München
Winter Semester 2021/22

Stein manifolds (4+2)

Winter Semester 2021/22

Description

Lecture notes

References

Problem session

Examination

 

1. Description

Stein manifolds are an important subject from complex analysis, considered as the theory of several complex variables. From the perspective of complex analysis the theory of Stein manifolds generalizes the theory of non-compact Riemann surfaces. While from the perspective of algebraic geometry, Stein manifolds correspond to affine schemes.

Alike to open Riemann surfaces, Stein manifolds are complex manifolds with many holomorphic functions. One can always find holomorphic functions with prescribed orders of zero at a discrete subset.

Today the investigation of Stein manifolds proceeds by using the language of sheaves and by proving vanishing results for their cohomology (Theorem B). These results are the point of comparison between Stein manifolds and affine schemes. It is planned to cover the following topics:

• Holomorphic functions of several variables
• Complex manifolds and sheaves
• Sheaf cohomology
• Coherent sheaves
• Cartan's lemma
• Theorem A and B on Stein manifolds
• Outlook

Target audience: Master mathematics or TMP. 9 ECTS (Module WP36, WP37)

Prerequisites: Complex analysis of a single variable (Funktionentheorie), Riemann surfaces, sheaf theory.

Scheduled time: Tuesday and Thursday 10-12 am (lecture),Thursday 12-2 pm (problem session).

It is planned to give the course online via Zoom. Please register by emailing to me. Then I will send you the link just before each Zoom session.

2. Lecture notes

Lecture notes

The lecture notes will be uploaded continuously during the course, always before the corresponding lecture.

3. References

Forster, Otto: Einführung in die Komplexe Analysis mehrerer Veränderlicher. Regensburger Trichter (1973)

Kaup, Ludger; Kaup, Burchard: Holomorphic Functions of Several Variables. An Introduction to the Fundamental Theory. de Gruyter (1983)

Forstnerič, Frank: Stein Manifolds and Holomorphic Maps. The Homotopy Principle in Complex Analysis. Springer 2nd ed. (2017)

Fritzsche, Klaus; Grauert, Hans: From holomorphic functions to complex manifolds. Springer (2002)

Grauert, Hans; Fritzsche, Klaus: Einführung in die Funktionentheorie mehrerer Veränderlicher. Springer (1977)

Grauert, Hans; Remmert, Reinhold: Theorie der Steinschen Räume. Springer (1977)

Hartshorne, Robin: Algebraic Geometry. Springer (1977)

4. Problem session

In addition to the lecture in class, each week a problem session will be held. The basis is a series of problems for homework. The students are encouraged to discuss their solutions during the problem session.

Problem sheets

5. Examination

To obtain the ECTS-Credits the participants have to pass the examination ("Modulprüfung") at the end of the course. The examination takes place as an oral examination via Zoom.

The examination will cover the topics from the lecture and from the problem sheets.

The examination is scheduled for Thursday, 24.2.2022.

It will take 30-60 minutes for each participant.

The use of written notes or electronic files or devices is not permitted during the examination.

If you want to attend the examination, please register by emailing to me until

Thursday, 3.2.2022,

with your name, student number, field of study, count of semester (and no. of the relevant module in case of master degree in mathematics).

Because the examination will be held via Zoom, in addition each participant has to fill in the file

Declaration of consent

Each participant is asked to send me a signed copy attached to his/her email.

I will send you a Zoom-Link according to the following schedule some minutes before your examination starts at 24.2.2022:

• Johannes Bartenschlager: 9.00 - 10.00 Uhr
• Mats Hansen: 10.00 - 11.00 Uhr
• Robin Mader: 11.00 - 12.00 Uhr
• Lukas Schönlinner: 13.00 - 14.00 Uhr
• Audrey Nkeng: 14.00 - 15.00 Uhr