  In the author's previous joint work with Hans-Joachim Hein, a mass formula
for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved,
assuming only relatively weak fall-off conditions on the metric. However, the
case of real dimension 4 presented technical difficulties that led us to
require fall-off conditions in this special dimension that are stronger than
the Chrusciel fall-off conditions that sufficed in higher dimensions. The
present article, however, shows that techniques of $4$-dimensional symplectic
geometry can be used to obtain all the major results of the previous paper,
assuming only Chrusciel-type fall-off. In particular, the present article
presents a new a proof of our Penrose-type inequality for the mass of an
asymptotically Euclidean Kaehler manifold that only requires Chrusciel metric
fall-off.

Differential Geometry