## Geometry & Topology

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

#### 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 $ℙ1$s 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
Revised: 2 March 2013
Accepted: 4 March 2013
First available in Project Euclid: 20 December 2017

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

#### 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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