Canonical metrics on four dimensional varieties

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The two main themes of the research proposed focus on the study of the moduli spaces of Einstein metrics and constant scalar curvature Kahler metrics at a point on the boundary of the classifying space. The first part of the proposal is a natural development of the PI's previous work, and plans to generalize aspects of Seiberg-Witten Theory, estimates on Riemannian curvatures, and obstructions to the existence of Einstein metrics on 4-orbifolds. Further implications of obstructions to the existence of Einstein metrics on smooth manifolds are expected. The other part the proposal addresses the study of Kahler structures. The project proposed on the classification of Ricci flat asymptotically locally euclidean Kahler manifolds (non-hyperkahler) establishes a starting point in the study of the compactification of the moduli space of constant scalar curvature Kahler manifolds. The specific tools used come from twistor spaces, geometric analysis, algebraic geometric stability, and Seiberg-Witten theory.In spite of the tremendous developments over the last twenty years, there is still a lot that eludes us about the geometry of manifolds in low dimensions. The proposed research aims to develop interactions between different geometric structures such as Riemannian geometry, differential topology and Kahler geometry on four dimensional spaces. The PI plans to approach classification problems of special geometries (ALE Ricci flat Kahler), and also to understand existence and non-existence of Einstein or constant scalar curvature Kahler geometric structures on 4-manifolds, or 4-manifolds with singularities. These topics are of great importance not only in differential and algebraic geometry, but also in mathematical physics (string theory).