## Journal of Differential Geometry

### Asymptotically cylindrical Calabi–Yau manifolds

#### Abstract

Let $M$ be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ 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 $\overline{M} \setminus \overline{D}$.

#### Article information

Source
J. Differential Geom., Volume 101, Number 2 (2015), 213-265.

Dates
First available in Project Euclid: 16 September 2015

https://projecteuclid.org/euclid.jdg/1442364651

Digital Object Identifier
doi:10.4310/jdg/1442364651

Mathematical Reviews number (MathSciNet)
MR3399097

Zentralblatt MATH identifier
1332.32028

#### Citation

Haskins, Mark; Hein, Hans-Joachim; Nordström, Johannes. Asymptotically cylindrical Calabi–Yau manifolds. J. Differential Geom. 101 (2015), no. 2, 213--265. doi:10.4310/jdg/1442364651. https://projecteuclid.org/euclid.jdg/1442364651