### Abstract

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_{M¯}| 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 language | English |
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Pages (from-to) | 213-265 |

Number of pages | 53 |

Journal | Journal of Differential Geometry |

Volume | 101 |

Issue number | 2 |

Early online date | 16 Sep 2015 |

DOIs | |

Publication status | Published - 31 Oct 2015 |

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### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

*101*(2), 213-265. https://doi.org/10.4310/jdg/1442364651