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

TL;DR

This paper explores the complex structures of hyperk"ahler four-manifolds with infinite topological type from the Gibbons-Hawking ansatz, revealing their biholomorphic equivalences to hypersurfaces or resolutions in complex three-space.

Contribution

It extends LeBrun's work by classifying complex structures of infinite topological type hyperk"ahler manifolds, showing their biholomorphic models in complex geometry.

Findings

Most complex structures are biholomorphic to hypersurfaces in c^3.

Remaining structures are biholomorphic to minimal resolutions of singular surfaces.

Partial extension of LeBrun's classification to countably punctured cases.

Abstract

In this paper, we study the complex structures of complete hyperk\"ahler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. We show that for almost all complex structures in the hyperk\"ahler 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.