Canonical metrics on four dimensional manifolds, and orbifold structures

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AbstractAward: DMS 1309029, Principal Investigator: Ioana Suvaina The main goal of the proposed research is the study of Riemannian geometry of orbifold spaces in dimension four, and how they appear as boundary points of the space of canonical metrics on smooth manifolds. The principal investigator proposes to analyze a family of problems addressing the study of the space of constant scalar curvature Kaehler surfaces. The projects proposed concern both problems on the local development of singularities, but also the understanding of compact smooth spaces. The principal investigator aims to establish connections between the algebraic geometry aspects of the problems and their counterparts in Kaehler geometry. The resolution of these projects will provide constructions of new examples of constant scalar curvature Kaehler surfaces, or constant scalar curvature almost Kaehler 4-manifolds. In another direction, the principal investigator proposes a series of projects on the relations between estimates of the curvature components, the topological and differential invariants of a 4-orbifold, and existence or non-existence of Einstein metrics. The main approach is via the Seiberg-Witten theory.The study of the four dimensional spaces has a long history both in mathematics and physics being motivated by the study of the space-time model. In mathematics, there is strong evidence that the four dimensional spaces are very special as they admit infinitely many smooth structures. This research program plans to use techniques from different fields in mathematics: differential topology, Riemannian and Kaehler geometry, and algebraic geometry to study four dimensional spaces. In particular, the principal investigator aims to establish new connections among these various fields.