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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {C}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> under certain conditions. Thus, we partially extend LeBrun’s celebrated work [LeBrun, C.: Complete Ricci-flat Kähler metrics on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {C} ^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> need not be flat. In Proc. Symp. Pure Math <b>52</b>, 297–304 (1991)] to the context of countably many punctures.</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月15日

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  • Wenxin He,
  • Bin Xu

摘要

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 \(\mathbb {C}^3\) 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 \(\mathbb {C}^3\) C 3 under certain conditions. Thus, we partially extend LeBrun’s celebrated work [LeBrun, C.: Complete Ricci-flat Kähler metrics on \(\mathbb {C} ^n\) C n need not be flat. In Proc. Symp. Pure Math 52, 297–304 (1991)] to the context of countably many punctures.