Geometry & Topology

Asymptotically cylindrical Calabi–Yau $3$–folds from weak Fano $3$–folds

Alessio Corti, Mark Haskins, Johannes Nordström, and Tommaso Pacini

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Abstract

We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–folds starting with (almost) any deformation family of smooth weak Fano 3–folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. Semi-Fano 3–folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3–folds, but are far more numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain 1s with normal bundle O(1)O(1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi–Yau 3–folds.

We introduce some general methods to compute the basic topological invariants of ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples.

All the features of the ACyl Calabi–Yau 3–folds studied here find application in [arXiv:1207.4470] where we construct many new compact G2–manifolds using Kovalev’s twisted connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds are particularly well-adapted for this purpose.

Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 1955-2059.

Dates
Received: 24 August 2012
Revised: 2 March 2013
Accepted: 4 March 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732646

Digital Object Identifier
doi:10.2140/gt.2013.17.1955

Mathematical Reviews number (MathSciNet)
MR3109862

Zentralblatt MATH identifier
1273.14081

Subjects
Primary: 14J30: $3$-folds [See also 32Q25] 53C29: Issues of holonomy
Secondary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14J28: $K3$ surfaces and Enriques surfaces 14J32: Calabi-Yau manifolds 14J45: Fano varieties 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords
differential geometry Einstein and Ricci-flat manifolds special and exceptional holonomy noncompact Calabi–Yau manifolds compact $G_2$–manifolds Fano and weak Fano varieties lattice polarised K3 surfaces

Citation

Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso. Asymptotically cylindrical Calabi–Yau $3$–folds from weak Fano $3$–folds. Geom. Topol. 17 (2013), no. 4, 1955--2059. doi:10.2140/gt.2013.17.1955. https://projecteuclid.org/euclid.gt/1513732646


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