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

By Alessio Corti, M. Haskins, Johannes Nordstrom, T. Pacini. 2012

108 citations recorded by the source corpus.

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 ℙ1s 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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fb530bc2471fd861361430f72d70d5277ca92ef4

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In Geometry and Topology

Citations in the corpus

G2-manifolds and associative submanifolds via semi-Fano 3-folds (2012, methodology): “We explain below in more detail how together with [18] and [37] this paper addresses problems with both steps (i) and (ii) in Kovalev’s original paper [44]—and therefore puts the twisted connected sum construction on a firm foundation—and also extends substantially the settings in which solutions to”
Notes on the octonions (2010, methodology): “There are however a number of construction techniques (all based on gluing methods): Joyce’s generalized Kummer construction for G2– and Spin(7)–manifolds [20, 21, 18, 19] based on resolving orbifolds of the form T7/Γ and T8/Γ; a method of Joyce’s for constructing Spin(7)–manifolds from real singula”
G2-instantons over twisted connected sums (2013, background): “The last two conditions in the definition of a building block are not essential; they have been made to facilitate the computation of certain topological invariants in [3].”
Asymptotically cylindrical Calabi-Yau manifolds (2012, methodology): “In [10] so-called weak Fano manifolds are used instead; the weak Fano construction yields hundreds of thousands of families of split ACyl Calabi-Yau 3-folds.”