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Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds. (English) Zbl 1273.14081

Summary: 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 \(\mathbb P^1\)s with normal bundle \(\mathcal O( - 1) \oplus \mathcal O( - 1)\), 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 the authors’ paper [“\(G_{2}\)-manifolds and associative submanifolds via semi-Fano 3-folds”, arXiv:1207.4470] where we construct many new compact \(G_{2}\)-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.

MSC:

14J30 \(3\)-folds
53C29 Issues of holonomy in differential geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J45 Fano varieties
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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