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Desingularizations of Conformally Kaehler, Einstein Orbifolds

Claude LeBrun, Tristan Ozuch

TL;DR

This work analyzes when limits of smooth Einstein 4-manifolds to Hermitian Einstein orbifolds with type-T singularities must, at large indices, be Kähler-Einstein, and that the limit is among Odaka–Spotti–Sun’s classified Kähler-Einstein orbifolds. It develops a robust gluing-desingularization framework (naïve desingularizations) to model how orbifold singularities are resolved by ALE gravitational instantons, underpinned by Wu’s criterion for conformally Kähler-Einstein metrics. By carefully controlling obstructions and the self-dual Weyl curvature, the authors show that admissible degenerations force the smooth manifolds to carry Kähler-Einstein structures, yielding strong topological and algebro-geometric constraints that sharply limit possible singular limits. The results imply that certain orbifold limits cannot arise from smooth Einstein manifolds, while offering a pathway to classify all admissible limits within the Odaka–Spotti–Sun program, and highlighting the central role of Type-T singularities and anti-self-dual bubbles in four-dimensional geometry.

Abstract

Let {(M,g_j)} be a sequence of smooth compact oriented Einstein 4-manifolds of fixed Einstein constant $λ> 0$ 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 j, the given (M,g_j) 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.

Desingularizations of Conformally Kaehler, Einstein Orbifolds

TL;DR

This work analyzes when limits of smooth Einstein 4-manifolds to Hermitian Einstein orbifolds with type-T singularities must, at large indices, be Kähler-Einstein, and that the limit is among Odaka–Spotti–Sun’s classified Kähler-Einstein orbifolds. It develops a robust gluing-desingularization framework (naïve desingularizations) to model how orbifold singularities are resolved by ALE gravitational instantons, underpinned by Wu’s criterion for conformally Kähler-Einstein metrics. By carefully controlling obstructions and the self-dual Weyl curvature, the authors show that admissible degenerations force the smooth manifolds to carry Kähler-Einstein structures, yielding strong topological and algebro-geometric constraints that sharply limit possible singular limits. The results imply that certain orbifold limits cannot arise from smooth Einstein manifolds, while offering a pathway to classify all admissible limits within the Odaka–Spotti–Sun program, and highlighting the central role of Type-T singularities and anti-self-dual bubbles in four-dimensional geometry.

Abstract

Let {(M,g_j)} be a sequence of smooth compact oriented Einstein 4-manifolds of fixed Einstein constant 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 j, the given (M,g_j) 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.
Paper Structure (14 sections, 26 theorems, 79 equations)

This paper contains 14 sections, 26 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Singularities of Type T
  3. Hermitian, Einstein Orbifolds
  4. Constraints on Diffeotype
  5. Admissible Limits and Desingularizations
  6. The Role of Wu's Criterion
  7. Desingularizing Einstein Orbifolds
  8. Gluing of ALE spaces to orbifold singularities.
  9. Simplest Case of the Proof
  10. Without Trees of Singularities
  11. For $k=0$,
  12. Now let $0\leqslant k \leqslant n-1$.
  13. The General Case
  14. Function Spaces

Key Result

Theorem A

Let $(X^4 , g_\infty)$ be a compact Kähler-Einstein orbifold of Einstein constant $\lambda >0$, and let $(M^4, g_i)$ be an admissible sequence of compact oriented Einstein manifolds with $(X , g_\infty)$ as its orbifold limit. Then, for all $i \gg 0$, the $(M, g_i)$ are actually all Kähler-Einstein.

Theorems & Definitions (58)

  • Definition 1
  • Theorem A
  • Theorem B
  • Definition 2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 48 more