CAREER: Canonical metrics and stability in complex geometry

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AbstractAward: DMS 1350696, Principal Investigator: Gabor SzekelyhidiGeometric partial differential equations govern much of the physical world. For example solutions of Einstein's equations are intimately related to our understanding of the universe. The proposed research studies differential equations related to Einstein's equations and the key question is how the global structure of a space influences the local, analytic properties, such as singularities of the solutions of such equations. Understanding this phenomenon will have applications in physics and the sciences in general. In addition the principal investigator proposes integrated research and educational programs. These programs will raise interest in high school students towards mathematics, and it will encourage undergraduate students to pursue K-12 STEM education.The principal investigator's proposed research is concerned with the study of canonical Kahler metrics on algebraic varieties, in particular the extremal metrics introduced by Calabi in the 1980's, and questions related to the J-flow introduced by Donaldson. The main conjecture in the field relates the existence of extremal metrics to the stability of the variety in the sense of geometric invariant theory, and the PI proposes an analogous conjecture for the J-flow. A particular feature of the proposal is a focus on situations when no extremal metric exists. On the algebro-geometric side the goal is to construct canonical degenerations of the variety and the PI's proposal is to use filtrations of the homogeneous coordinate ring, in analogy with Harder-Narasimhan filtrations of unstable vector bundles. On the differential geometric side one needs to understand the possible limiting behavior of families of extremal metrics. In general this is much more intricate than the much more thoroughly understood case of Kahler-Einstein metrics, and the PI proposes to first restrict attention to metrics with bounded curvature, and to relate the limiting behavior to filtrations. In the proposal a special emphasis is placed on the construction of new examples of extremal metrics, and the applications of these ideas to other problems in Kahler geometry.