Mass in K\"ahler Geometry

TL;DR

This paper derives an explicit formula for the mass of ALE K"ahler manifolds, showing it is a topological invariant in scalar-flat cases and establishing positivity and inequalities for the mass.

Contribution

It provides a simple, explicit mass formula for ALE K"ahler manifolds and proves the mass is a topological invariant in scalar-flat cases, with implications for positivity and inequalities.

Findings

Mass formula depends only on topology and complex structure.

Mass is a topological invariant for scalar-flat ALE K"ahler manifolds.

Positive mass theorem and Penrose inequality are established for ALE K"ahler metrics.

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.