The holomorphic symplectic structures on hyper-Kähler manifolds of type A∞

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Hyper-Kähler manifolds of type A∞ are noncompact complete Ricci-flat Kähler manifolds of complex dimension 2. We study the holomorphic symplectic structures preserved by the natural C×-actions on these manifolds, and we give sufficient and necessary conditions for the existence of C×-equivariant biholomorphisms between two hyper-Kähler manifolds of type A∞ preserving their holomorphic symplectic structures. As a consequence, we can show the existence of a complex manifold of dimension 2 on which there is a continuous family of complete Ricci-flat Kähler metrics with distinct volume growth.

Original languageEnglish
Pages (from-to)613-630
Number of pages18
JournalAdvances in Geometry
Issue number4
Publication statusPublished - 2014 Oct 1
Externally publishedYes


ASJC Scopus subject areas

  • Geometry and Topology

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