Mass in Kaehler Geometry

Abstract: Given a complete Riemannian manifold that looks enough like Euclidean space at infinity, physicists have defined a quantity called the “mass” that measures the asymptotic deviation of the geometry from the Euclidean model. This lecture will explain a simple formula, discovered in joint work with Hajo Hein, for the mass of any asymptotically locally Euclidean (ALE) Kaehler manifold. For ALE scalar-at Kaehler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kaehler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies the positive mass theorem for Kaehler metrics, but also yields a Penrose-type inequality for the mass. I will also briefly indicate some recent  technical improvements that allow one to prove these  results with only extremely weak metric fall-off hypotheses at infinity.