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