Complex structures of the Gibbons-Hawking ansatz with infinite topological type

In this paper, we study the complex structures of complete hyperkähler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. We show that for almost all complex structures in the hyperkähler family, the manifold is biholomorphic to a hypersurface in $\mathbb{C}^3$ defined by an explicit entire function. For the remaining complex structures, we further prove that the manifold is biholomorphic to the minimal resolution of a singular surface in $\mathbb{C}^3$ under certain conditions. Thus, we partially extend LeBrun’s celebrated work to the context of countably many punctures.

💡 Research Summary

The paper investigates the complex structures of complete hyperkähler four‑manifolds that arise from the Gibbons‑Hawking construction when the underlying harmonic function has countably many singularities, i.e., the manifold has infinite topological type. Starting from a discrete, closed, countable set of points A={p_j}⊂ℝ³, the authors consider a principal S¹‑bundle π₀:M₀→ℝ³\A whose first Chern class evaluates to –1 on a small sphere around each puncture. The Gibbons‑Hawking potential is taken to be

 V(p)=∑_{j≥1} 1/(2‖p−p_j‖),

which is well‑defined under the summability condition

 ∑_{j≥2} 1/‖p₁−p_j‖ < ∞.

With this V, the metric

 g = V⁻¹ ω² + V π₀* g_Euc

extends smoothly over the added S¹‑fixed points, yielding a complete hyperkähler manifold M of infinite topological type.

Existence criterion (Theorem 1.1).
The authors reformulate the condition for a positive harmonic function V to represent a given cohomology class e=(e₁,e₂,…)∈H²(U,ℤ) (U=ℝ³\A). Using the Riesz measure of the subharmonic function –V, they prove that such a V exists iff every e_j≤0 and there is a point x∈U with

 ∑_{j≥1} |e_j|/‖x−p_j‖ < ∞.

The necessity is shown via Bôcher’s theorem and a careful analysis of the Riesz mass at each puncture; sufficiency follows from a monotone convergence argument together with Harnack’s theorem. This provides a potential‑theoretic proof that differs from earlier maximum‑principle arguments.

Complex structures in the hyperkähler family.
For each unit vector v∈S² the hyperkähler family contains a complex structure

 J_v = v₁J_x + v₂J_y + v₃J_z,

where (J_x,J_y,J_z) are the standard quaternionic structures associated with the Gibbons‑Hawking ansatz. The authors project the puncture set onto the plane orthogonal to v via Π_v and denote the projected points by a_j(v). The distinct projected values are {b_k(v)} with multiplicities m_k(v)=#{j : a_j(v)=b_k(v)}.

Theorem 1.3 (generic directions).
If the punctures satisfy the same summability condition as above, then there exists a full‑measure subset B⊂S² such that for every v∈B the complex manifold (M,J_v) is biholomorphic to the smooth hypersurface in ℂ³ defined by

 u₁ u₂ = P_v(u₃),

where P_v is an explicit entire function determined by the projected points {a_j(v)}. The proof adapts LeBrun’s method for finitely many fixed points, replacing finite sums by convergent series and using potential theory to control the growth of P_v. Consequently, for almost every complex structure the hyperkähler manifold is algebraically realized as a single hypersurface in ℂ³.

Theorem 1.4 (exceptional directions).
For directions v∈S²\B the hypersurface description may fail. Under two additional finiteness hypotheses—(1) the set {b_k(v)} has no accumulation point in ℂ, and (2) each multiplicity m_k(v) is finite—the authors prove that (M,J_v) is biholomorphic to the minimal resolution of the singular hypersurface

 u₁ u₂ = P_v(u₃).

The singularities are of type A_{m_k(v)−1} and occur at the points where the projection has multiplicity greater than one. This extends LeBrun’s finite‑point classification to the countable‑point setting, showing that even the exceptional complex structures are still algebraically describable, albeit after resolving isolated quotient singularities.

Methodological highlights.

• The use of Riesz measures provides a clean bridge between cohomological data (the Chern class) and analytic data (the harmonic potential).
• Bôcher’s theorem is employed to rule out positive coefficients in the cohomology class, ensuring the negativity required for smooth extension.
• The projection Π_v and the associated multiplicity data translate the geometric configuration of punctures into the coefficients of the entire function P_v, making the algebraic model explicit.
• Measure‑theoretic arguments (full‑measure subset of S²) guarantee that the generic case covers “almost all” complex structures, while the finiteness conditions isolate the exceptional set.

Significance.
The paper demonstrates that hyperkähler 4‑manifolds with infinitely many S¹‑fixed points—previously known mainly through metric and topological properties—admit a remarkably concrete complex‑analytic description. Most complex structures are globally modeled by a single entire‑function hypersurface in ℂ³; the remaining ones are minimal resolutions of singular hypersurfaces with only A‑type quotient singularities. This bridges the gap between infinite‑dimensional hyperkähler quotient constructions and classical algebraic geometry, extending LeBrun’s finite‑type results to the infinite‑type realm and providing new tools for studying the interplay between topology, metric geometry, and complex structure in higher‑dimensional gauge‑theoretic settings.