Asymptotically cylindrical Calabi-Yau manifolds
Abstract
© 2015, International Press of Boston, Inc. All rights reserved. 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<inf>M¯</inf>| 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¯.
Type
Journal articlePermalink
https://hdl.handle.net/10161/19598Collections
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