A complete connected non-compact Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) is said to be asymptotically Euclidean (or AE) if there is a compact subset \(\mathbf{K}\subset M\) such that \(M-\mathbf{K}\) consists of finitely many components, each of which is diffeomorphic to the complement of a closed ball \(\mathbf{D}^n \subset \mathbb{R}^n\), in a manner such that \(g\) becomes the standard Euclidean metric plus terms that fall off sufficiently rapidly at infinity. The components of \(M-\mathbf{K}\) are called the ends of \(M\); More generally, \((M,g)\) is said to be asymptotically locally Euclidean (or ALE) if each end is diffeomorphic to the quotient \((\mathbb{R}^n-\mathbf{D}^n)/\Gamma_j\), where \(\Gamma_j\subset {\mathbf O}(n)\) is a finite subgroup which acts freely on the unit sphere. The mass of an ALE manifold is given by \[ \mathfrak{m}(M, g) := \lim_{\varrho\to \infty} \frac{\Gamma(\frac{n}{2})}{4(n-1)\pi^{n/2}} \int_{S_\varrho/\Gamma_j} \left[ g_{k\ell, k} -g_{kk,\ell}\right] \mathbf{n}^\ell d\mathfrak{a}_E \] where \(S_\varrho\) is the Euclidean coordinate sphere of radius \(\varrho\), \(d\mathfrak{a}_E\) is the \((n-1)\)-dimensional volume form on this sphere, and \(\vec{\mathbf{n}}\) is the outward-pointing Euclidean unit normal vector. The mass may be understood as an anomaly in the formula for the total scalar curvature, encapsulating an essential difference between the ALE and compact cases.
In this paper, the authors prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well defined. For ALE scalar-flat Kähler 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 Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), this formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.