Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

This proposal deals with the existence, moduli, and special properties of solutions of geometric differential equations (analogous to the Einstein 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 theorem for compact complex curves. In higher dimensions, the natural extension is the problem proposed by Calabi in the 1980's of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. There are some other related problems considered in the proposal, which naturally arise in complex geometry, and one recurrent theme is a strong intertwining of algebraic geometry, differential geometry, and global analysis, with each providing complementary insight into the structure of the spaces of solutions.

The specific objectives described in this proposal attempt to address the following directions:

(a) Find extremal Kaehler metrics on special polarized varieties, such as toric varieties whose Delzant polytopes are cuboids, or projective bundles over special varieties, by reducing the corresponding equations to simpler partial differential equations. Link the (non)existence of solutions in these special cases with various algebro-geometric notions of stability for the corresponding polarized varieties.

(b) Extend the theory to other geometric equations of current interest, such as the one corresponding to conformally-Kaehler, Einstein-Maxwell metrics.

(c) Find natural substitutes for extremal Kaehler metrics for non-Kahler compact complex surfaces.

The study of (internal or external) symmetries and some rather subtle separation of variables techniques will play an important role in achieving these objectives.