Mathematics

This is a follow-up to my course "Kaehler geometry" and is meant for those wishing to know more about (or possibly work on) the relations between stability notions for projective varieties and the existence of canonical metrics in Kaehler geometry. We will start with proving Donaldson's classical algebro-geometric lower bound for the Calabi functional and then proceed with the notion of K-stability, studying the space of test-configurations in some detail. Further topics will be decided taking into account the interests of the audience.

Kaehler geometry studies complex manifolds with a hermitian metric adapted to the complex structure. I plan to start from the basics and then concentrate on some aspects of the theory of canonical metrics on compact Kaehler manifolds (including the Calabi-Yau theorem). Some other topics (e.g. the hyperkaehler condition; variational approach to Kaehler-Einstein metrics; deformation theory of Kaehler-Einstein metrics) may be covered depending on the interests of the audience.

Some references:

The lectures will follow the book : "Complex Algebraic Surfaces" by Arnauld Beauville