Special metrics in classical and generalized Kaehler geometry

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This proposal deals with the existence, moduli, and special geometrical properties of solutions of geometric differential equations (analogous to Einstein's field equation in general relativity) naturally arising in the study of complex varieties. The prototype is the problem of existence and uniqueness of a metric of constant Gauss curvature on a compact complex curve, whose resolution leads to the famous uniformization of compact complex curves. In the 1980's, Calabi formulated a more general problem of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. From the point of view of global analysis, extremal Kaehler metrics are given by solutions to a non-linear 4th order PDE for which no general methods are currently available. Particular cases include Kaehler metrics of constant scalar curvature and Kaehler-Einstein metrics. The projects in this proposal attempt to address the following problems: (a) Find extremal metrics on special varieties by reducing the corresponding equations to simpler PDE's. Link the (non)existence of solutions in these special cases with various algebro-geometric notions of (un)stability for the corresponding varieties, thus providing a fertile testing ground for the main conjectures of the general theory. (b) Extend the notion of extremal Kaehler metric to generalized Kahler geometry and find non-trivial examples. The study of (internal or external) symmetries will plain an important role in achieving these objectives.