turkmath.org

An extremal Kähler metric is a canonical Kähler metric, introduced by E. Calabi, which is somewhat more general than a constant scalar curvature Kähler metric. The existence of such a metric is an ongoing research subject and expected to be equivalent to some form of geometric stability of the underlying polarized complex manifold $(M, J, [\omega])$ –the Yau-Tian-Donaldson conjecture. Thus it is no surprise that there is a moment map, the scalar curvature (A. Fujiki, S. Donaldson), and the problem can be described as an infinite dimensional version of the familiar finite dimensional G.I.T.

I will describe how the moment map can be used to describe the local space of extremal metrics on a symplectic manifold. Essentially, the local picture can be reduced to finite dimensional G.I.T. In particular, we can construct a course moduli space of extremal Kähler metrics with a fixed polarization $[\omega] \in  H^2(M, \mathbb{R})$, which is an Hausdorff complex analytic space