Mass in K\"ahler Geometry

Abstract

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) K\"ahler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat K\"ahler 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 K\"ahler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for K\"ahler metrics, but also yields a Penrose-type inequality for the mass.