# Seminar: Kähler manifolds

Prof. Bernd Ammann, Zimmer 119

## Content of the Seminar

Kähler manifolds are complex manifolds together with a compatible
Riemannian metric. Complex structures on a manifold M
are expressed by a section J∈End(TM) satisfying J^{2}=-1, and
compatibility with a metric means that ω(X,Y):=g(X,Y) defines
a symplectic struture on M. The simplest example is
C^{m} with the
standard complex structure and its standard metric. Thus the seminar
is intensively related both to symplectic geometry,
Riemannian geometry and complex varieties.
In Riemannian geometry Kähler manifolds are important as the complex structure
allows many calculations and as one can obtain many interesting examples.
As an example of the large achievements of Kähler geometry we mention
Yau's construction of compact
Ricci-flat Kähler manifolds, called Calabi-Yau manifolds, which are important
in many areas
including Riemannian geometry, physics and even in archimedean geometry.

The main focus of the seminar, is however, not to understand these advanced
topics, but to learn the basic techniques in the field: complex manifolds,
holomorphic bundles, (p,q)-forms, blowups, Kähler manifolds, Laplace
operators, and Hodge and Dolbeault theories.
Some of theses topics may also be the subject of a bachelor thesis, possibly in combination with the lecture on symplectic manifolds.

More advanced topics such as the construction of Calabi-Yau manifolds, the
proof of Kodaira's embedding theorem, the Kähler-Ricci flow or the
moduli spaces of Calabi Yau manifolds might be --- depending
on the participants --- sketched at the end of the seminar,
discussed in more detail in a
seminar in the summer term, or lead to a master thesis.

## Literature

- Andrei Moroianu, Lectures on Kähler geometry, London Math. Soc.,
Student texts 69
- Werner Ballmann, Lectures on Kähler geometry, ESI Lectures in
Mathematics and Physics, EMS, available in Regensburg as Ebook.

## Requirements

For participation it is helpful to have followed an introductory
lecture into differential geometry. It is also recommended
to follow my lecture on symplectic manifolds during the winter term.
We will not use many results from these lectures, but some experience
with manifolds and vector bundles is important for understanding the subject.
## Time and Place

Monday 16-18, M102.

## Program

Program of the seminar.
## Registration

Presentation of the topic and distribution of talks on
**August 1, 2017, 13:30**
in the Sitzungszimmer Mathematik (M201). After this meeting please
register via Email to Bernd Ammann.

Please also register in G.R.I.P.S., as we send announcements
via this system.
## Related web pages

## Formal requirements (in German)

### Kriterien für benotete Leistungsnachweise

Um die üblichen Leistungsnachweise zu erhalten, sind folgende
Kriterien zu erfüllen:
- Erfolgreiches Vortragen
- Schriftliche Ausarbeitung eines Vortrages
- aktive Mitarbeit im Seminar

### Unbenotete Leistungsnachweise

Wie bei benoteten Leistungsnachweisen.
### Modulteilprüfung

Vortrag und Ausarbeitung bilden die Modulteilprüfung.
Bernd Ammann, 10.07.2017 oder später