**Title:** Donaldson Theory

**Speaker:** Steve Zelditch

**Speaker Info:**

**Brief Description:**

**Special Note**:

**Abstract:**

The problem I will discuss is whether there exists a CSC (constant scalar curvature) metric in each ``Kaehler class" of a compact Kaehler manifold. This is the higher dimensional analogue of the uniformization theorem for Riemann surfaces. But there exist Kaehler manifolds for which there exists no CSC metric in the Kaehler class.A long running program of Yau-Tian-Donaldson (and others such as Mabuchi, Futaki, Phong...) is to find a necessary and sufficient condition for existence of a CSC metric in terms of a GIT (= geometric invariant theory) notion of stability. It is perhaps the principal problem in complex geometry, and the main subject of the upcoming Donaldson conference.The stability notion is part of infinite dimensional GIT theory.

In the second talk, I will concentrate on infinite dimensional GIT theory. I will review finite dimensional GIT stability and then explain the infinite dimensional generalization. It turns out that the scalar curvature is the moment map for the action of the group of symplectic diffeos of a Kaehler manifold acting the on space of compatible complex structures. Finding a zero of the moment map is finding a CSC metric in a class. It should be equivalent to stability of the class in the sense that the Mabuchi K-energy is proper on it. To do infinite dimensional GIT, it is very useful to make finite dimensional approximations by means of Bergman kernels and Bergman metrics.

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