[Paper Review] Desingularisation de metriques d'Einstein. II

Olivier Biquard|arXiv (Cornell University)|Nov 4, 2013

Geometry and complex manifolds20 references25 citations

TL;DR

This paper establishes a wall-crossing formula for 4-dimensional Poincaré-Einstein metrics by analyzing the desingularization of Einstein metrics with A1 orbifold singularities. Using a formalism based on hyperkähler geometry and curvature decomposition in terms of self-dual 2-forms, it computes second-order terms in the Einstein equation to determine the sign of the determinant of the self-dual curvature operator at the singular point. The key result is that Einstein desingularizations exist precisely on the side of the moduli wall where det R+(g0(γ))(p0) > 0, resolving a question from prior work and extending to Dk and Ek singularities via higher-order jet obstructions.

ABSTRACT

We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.

Motivation & Objective

• To determine the side of the moduli wall in the conformal infinity space where Einstein desingularizations exist for singular 4-dimensional Einstein metrics with A1 orbifold singularities.
• To develop a general formalism for computing second-order terms in the Einstein equation using SO(3)-connections on self-dual 2-forms.
• To extend the desingularization procedure to higher-order singularities (Dk, Ek) by analyzing obstructions from higher jets of the metric at the singular point.
• To resolve the sign ambiguity in the wall-crossing behavior by computing the sign of det R+(g0(γ))(p0) for the desingularized metrics.
• To identify the first non-trivial obstruction to desingularization beyond the initial condition (1), specifically at order 4 for Ak singularities.

Proposed method

• Uses a formalism based on SO(3)-connections on the bundle of self-dual 2-forms to decompose the curvature tensor and express the Einstein equation in terms of curvature components R+ and R−.
• Applies the Gibbons-Hawking ansatz to construct explicit hyperkähler metrics on A1, Dk, and Ek singularities, enabling explicit computation of curvature and connection terms.
• Computes second-order terms in the Einstein equation by analyzing the variation of the self-dual curvature operator R+ under deformation of the conformal infinity.
• Uses the moment map and holomorphic 1-forms on the hyperkähler manifold to compute integrals over the desingularizing sphere, particularly ∫Σ m ω1 and ∫Σ ϕ1.
• Applies the formalism to the case of Ak singularities by computing the 2-jet and 4-jet of the metric at the singular point, identifying obstructions to desingularization.
• Establishes that for Dk and Ek singularities, the obstruction at order 4 vanishes due to symmetry, suggesting higher-order obstructions (e.g., order 6) may arise.

Experimental results

Research questions

• RQ1 On which side of the moduli wall C0 in the space of conformal boundaries does the desingularized Einstein metric exist?
• RQ2 What is the precise sign condition on det R+(g0(γ))(p0) that determines the existence of a desingularized Einstein metric?
• RQ3 How do higher-order jet obstructions (beyond the 2-jet) affect the desingularization process for Ak singularities?
• RQ4 Can the wall-crossing behavior be generalized from A1 to Dk and Ek singularities using the same curvature-based formalism?
• RQ5 What is the role of the hyperkähler geometry of gravitational instantons in computing the second-order terms of the Einstein equation?

Key findings

• The desingularized Einstein metrics exist precisely on the side of the moduli wall where det R+(g0(γ))(p0) > 0, resolving the sign ambiguity in the wall-crossing formula.
• For A1 singularities, the obstruction to desingularization at order 4 vanishes if the 4-jet of the metric satisfies a symmetry condition, allowing the use of conformal deformations to resolve the determinant condition.
• In the Dk and Ek cases, the obstruction at order 4 vanishes due to the C*-action fixing three singular points, implying that the obstruction arises at higher order (likely order 6).
• The formalism successfully computes second-order terms in the Einstein equation using the SO(3)-connection and curvature decomposition, enabling explicit analysis of the desingularization process.
• The computation of ∫Σ m ω1 = π(k+1)(Vol Σ / 2π)^2 confirms the geometric origin of the obstruction and validates the ansatz for Ak instantons.
• The paper shows that the condition det R+(g0)(p0) = 0 is necessary for desingularization, and that the sign of det R+(g0(γ))(p0) determines the existence of solutions on one side of the wall.