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, [ω]) –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.

In this talk I will give an introduction to extremal metrics. Then I will describe how the moment map can be used to describe the local deformation problem of extremal metrics. 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 [ω] ∈ H2(M, R), which is an Hausdorff complex analytic space.