Desingularizations of Conformally Kaehler, Einstein Orbifolds

TL;DR

This paper proves that under certain conditions, sequences of Einstein 4-manifolds converging to a Hermitian orbifold are eventually Kähler-Einstein, linking the limit's geometry to known classifications.

Contribution

It establishes that sequences of Einstein 4-manifolds with Hermitian limits and anti-self-dual bubbling are eventually Kähler-Einstein, connecting convergence behavior to orbifold classifications.

Findings

Sequences become Kähler-Einstein for large indices.

Limit orbifold is Kähler-Einstein and classified by prior work.

Bubbling instantons are anti-self-dual.

Abstract

Let {(M,g_i)} be a sequence of smooth compact oriented Einstein 4-manifolds of fixed Einstein constant $\lambda > 0$ that Gromov-Hausdorff converges to a 4-dimensional Einstein orbifold X. Suppose, moreover, that the limit metric is Hermitian with respect to some complex structure on the limit orbifold X, that X has at least one singular point, and that every gravitational instanton that bubbles off from the sequence is anti-self-dual. Then, for all sufficiently large i, the given (M,g_i) are all Kaehler-Einstein. As a consequence, the limit orbifold X is also Kaehler-Einstein, and must in fact be one of the orbifold limits classified by Odaka, Spotti, and Sun.