# Asymptotically cylindrical Calabi-Yau manifolds

Journal Article (Journal Article)

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¯ ∈ |-KM¯| 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¯.

### Full Text

### Duke Authors

### Cited Authors

- Mark, H; Hein, HJ; Johannes, N

### Published Date

- October 1, 2015

### Published In

### Volume / Issue

- 101 / 2

### Start / End Page

- 213 - 265

### Electronic International Standard Serial Number (EISSN)

- 1945-743X

### International Standard Serial Number (ISSN)

- 0022-040X

### Digital Object Identifier (DOI)

- 10.4310/jdg/1442364651

### Citation Source

- Scopus