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

Abstract: In this paper, we study the complex structures of complete hyperkähler four-manifolds of infinite topological type arising from the Gibbons-Hawking ansatz. Under a natural genericity assumption on the monopole configuration, we show that the resulting manifold is biholomorphic to a hypersurface in $\mathbb{C}<sup>3$ defined by an explicit entire function, and we prove that this description persists for almost all complex structures in the associated $S<sup>2$-family of compatible complex structures. When the genericity condition is relaxed, we further demonstrate that the manifold is biholomorphic to the minimal resolution of a singular surface. Thus, we partially extend LeBrun's celebrated work to the context of countably many monopoles.