WiSe 14/15 - Lecture course

Differential geometry of complex vector bundles

Time and place:

Tue, 10 - 12, WSC-O-4.65
Fr, 10 - 12, WSC-S-U-3.03

Exercise session

Tue, 12 - 14, WSC-S-U-4.02

LSF-Entry

here

First lecture

Fr, October 17th

First exercise session

Tue, November 4th

News

  • On Tuesday, Dec. 2nd, there will be no lecture. The lecture of Friday, Dec. 5th, will be moved to Thursday, Dec. 4th, noon until 2 pm. The lecture will take place in WSC-O-3.46.
  • Starting with Problem Set 2, the homework is due on Fridays, at the end of the lecture.
  • There will be no lectures on October 31st and November 4th. The exercise session will start on November 4th (12 - 14 h in WSC-S-U-4.02). There will be an additional exercise session on November 7th, which will take place instead of the Friday lecture at the usual time and place.
  • On Friday, October 24th, the course will take place from 8.30 - 10.00 in WSC-S-3.14.
  • Starting on Tue, October 28th, the Tuesday session of the course will take place in WSC-O-4.65.


Description

The aim of this lecture is to give a first introduction into Complex differential geometry, with a special emphasis on vector bundles. It is aimed at students with a basic knowledge of Algebraic Geometry that want to learn more about vector bundles from a differential geometric viewpoint. We will introduce the basic tools of complex differential geometry and try to understand how they relate to concepts in Algebraic Geometry such as characteristic classes, ampleness and stability.

Preliminary list of contents:

complex manifolds and holomorphic vector bundles, Kähler manifolds, Lie groups, principal bundles, connections in principal bundles, characteristic classes via curvature forms, especially Chern classes, if time permits: self-dual connections and relations to algebro-geometric stability.

References

  • Demailly: Complex analytic and differential geometry, book available from the author's website: download here
  • Huybrechts: Complex Geometry, Springer
  • Moroianu: Lectures on Kähler Geometry, Cambridge University Press
  • Voisin: Hodge Theory and Complex Algebraic Geometry I, Cambridge University Press
  • Wells: Differential Analysis on Complex Manifolds, Springer

Recap on differentiable manifolds and differential forms

In case you want to read up the basics on differentiable manifolds and differential forms, I recommend for example Sections 1.1, 1.2, 2.1, 2.2, 3.1, 3.2, 3.4 of Moroianu's book (less than 15 pages in total), preliminary versions of which should also be available on the web.

Problem sheets