Asymptotically cylindrical Calabi-Yau manifolds

Haskins Mark, Hans Joachim Hein, Nordström Johannes

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Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) x X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M) = SU(n), where n is the complex dimension of M. If n > 2 we then show that there exists a projective orbifold M¯ and a divisor D¯ ∈ |-K| with torsion normal bundle such that M is biholomorphic to M¯ \ D¯, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M¯ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M¯, D¯) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M¯ \ D¯.

Original languageEnglish
Pages (from-to)213-265
Number of pages53
JournalJournal of Differential Geometry
Issue number2
Early online date16 Sept 2015
Publication statusPublished - 31 Oct 2015

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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