Open Access
October 2015 Asymptotically cylindrical Calabi–Yau manifolds
Mark Haskins, Hans-Joachim Hein, Johannes Nordström
J. Differential Geom. 101(2): 213-265 (October 2015). DOI: 10.4310/jdg/1442364651


Let $M$ be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ 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 $\mathrm{Hol}(M) = \mathrm{SU}(n)$, where $n$ is the complex dimension of $M$. If $n \gt 2$ we then show that there exists a projective orbifold $\overline{M}$ and a divisor $\overline{D} \in \lvert -K_{\overline{M}} \rvert$ with torsion normal bundle such that $\overline{M}$ is biholomorphic to $\overline{M} \setminus \overline{D}$, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting.We give examples where $\overline{M}$ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair $\overline{M} \setminus \overline{D}$ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi–Yau metrics on $\overline{M} \setminus \overline{D}$.


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Mark Haskins. Hans-Joachim Hein. Johannes Nordström. "Asymptotically cylindrical Calabi–Yau manifolds." J. Differential Geom. 101 (2) 213 - 265, October 2015.


Published: October 2015
First available in Project Euclid: 16 September 2015

zbMATH: 1332.32028
MathSciNet: MR3399097
Digital Object Identifier: 10.4310/jdg/1442364651

Rights: Copyright © 2015 Lehigh University

Vol.101 • No. 2 • October 2015
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