[Paper Review] A sm\"org {a}sbord of scalar-flat K\"ahler ALE surfaces

Michael T. Lock, Jeff Viaclovsky|arXiv (Cornell University)|Oct 23, 2014

Geometry and complex manifolds38 references3 citations

TL;DR

This paper constructs scalar-flat Kähler ALE metrics on the minimal resolution of ℂ²/Γ for any non-cyclic finite subgroup Γ⊂U(2) without complex reflections, proving existence via holomorphic circle actions and deformation techniques. The key result shows such metrics exist for all such Γ, including non-SU(2) groups, and that they are hyperkähler if and only if Γ⊂SU(2).

ABSTRACT

There are many known examples of scalar-flat K\\"ahler ALE surfaces, all of which have group at infinity either cyclic or contained in ${\ m{SU}}(2)$. The main result in this paper shows that for any non-cyclic finite subgroup $\\Gamma \\subset {\ m{U}}(2)$ containing no complex reflections, there exist scalar-flat K\\"ahler ALE metrics on the minimal resolution of $\\mathbb{C}^2 / \\Gamma$, for which $\\Gamma$ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat K\\"ahler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal K\\"ahler metrics admitting holomorphic isometric circle actions in certain K\\"ahler classes on the complex analytic compactifications of the minimal resolutions.

Motivation & Objective

• To extend the known existence of scalar-flat Kähler ALE metrics beyond cyclic and SU(2)-type groups to all non-cyclic finite subgroups of U(2) without complex reflections.
• To establish the existence of such metrics on the minimal resolution of ℂ²/Γ with Γ as the group at infinity.
• To show these metrics admit a holomorphic isometric circle action, linking geometry to symmetry.
• To demonstrate existence for small deformations of the complex structure of the minimal resolution.
• To prove the existence of extremal Kähler metrics with holomorphic circle actions on compactifications of the minimal resolutions.

Proposed method

• Utilizes the scalar-flat Kähler gluing theorem from Section 5.1 to construct metrics by attaching multi-Eguchi-Hanson metrics to quotient spaces.
• Applies Theorem 8.5 on self-dual orbifold connect sums to glue conformal compactifications of scalar-flat anti-self-dual ALE metrics.
• Employs conformal blow-up at orbifold points of positive Yamabe invariant compactifications to produce scalar-flat anti-self-dual ALE metrics.
• Uses the eta invariant η(S³/Γ) and b₂⁻(X̃) in Lemma 8.1 to test Ricci-flatness and conclude hyperkähler structure when Γ⊂SU(2).
• Analyzes the exceptional divisor as a tree of rational curves with three Hirzebruch-Jung strings attached to a central curve with self-intersection −bΓ.
• Relies on Brieskorn’s classification of quotient singularities and the correspondence between Γ and the resolution graph ⟨−bΓ; (α₁,β₁); (α₂,β₂); (α₃,β₃)⟩.

Experimental results

Research questions

• RQ1 Can scalar-flat Kähler ALE metrics be constructed for all non-cyclic finite subgroups Γ⊂U(2) without complex reflections?
• RQ2 Do these metrics on the minimal resolution of ℂ²/Γ admit a holomorphic isometric circle action?
• RQ3 Do such metrics exist for small deformations of the complex structure of the minimal resolution?
• RQ4 Are there extremal Kähler metrics with holomorphic circle actions on the complex analytic compactifications of the minimal resolutions?
• RQ5 When is such a scalar-flat Kähler ALE metric necessarily hyperkähler?

Key findings

• For any non-cyclic finite subgroup Γ⊂U(2) with no complex reflections, scalar-flat Kähler ALE metrics exist on the minimal resolution of ℂ²/Γ with Γ as the group at infinity.
• These metrics admit a holomorphic isometric circle action, providing a symmetry structure on the resolution.
• Scalar-flat Kähler ALE metrics exist for small deformations of the complex structure of the minimal resolution, extending the moduli space of such metrics.
• Extremal Kähler metrics with holomorphic isometric circle actions exist in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.
• The metrics are hyperkähler if and only if Γ⊂SU(2), as determined by the inequality in Lemma 8.1 using η(S³/Γ) and b₂⁻(X̃).
• The only building blocks required in the construction are multi-Eguchi-Hanson metrics, confirming their foundational role in the gluing process.