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 J2=-1, and compatibility with a metric means that ω(X,Y):=g(X,Y) defines a symplectic struture on M. The simplest example is Cm 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.



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 of the seminar.


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:
  1. Erfolgreiches Vortragen
  2. Schriftliche Ausarbeitung eines Vortrages
  3. aktive Mitarbeit im Seminar

Unbenotete Leistungsnachweise

Wie bei benoteten Leistungsnachweisen.


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