Teachers: Vincent Koziarz and Andrea Fanelli
Program:
In this course, we will deal with higher dimensional complex geometry, with a focus on Kähler manifolds. This course is complementary to the course of algebraic geometry, but we will give a preference to transcendental methods.
The following topics will be discussed
- Introduction to holomorphic functions of several variables
- Complex manifolds, de Rham and Dolbeault cohomology
- Sheaves and cohomology
- Vector bundles, connections and curvature
- Harmonic theory on compact complex manifolds, Hodge theory
- Kähler manifolds, Hodge decomposition
- Line bundles, first Chern class, notions of positivity for the curvature, Kodaira vanishing theorem
- Kodaira embedding theorem
Bibliography:
J.-P. Demailly: Complex analytic and algebraic geometry, available on the web page of J.-P. Demailly.
P. Griffiths et J. Harris: Principles of algebraic geometry, Wiley & Sons, 1978
S. Kobayashi: Differential geometry of complex vector bundles, Princeton University Press, 1987
R.O. Wells: Differential analysis on complex manifolds, GTM 65, Springer, 1980