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

Alessio Corti, Mark Haskins, Johannes Nordström and Tommaso Pacini

Geometry & Topology 17 (2013) 1955–2059

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.

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
Mathematical Subject Classification 2000
Primary: 14J30, 53C29
Secondary: 14E15, 14J28, 14J32, 14J45, 53C25
Received: 24 August 2012
Revised: 2 March 2013
Accepted: 4 March 2013
Published: 15 July 2013
Proposed: Richard Thomas
Seconded: Ronald Fintushel, Yasha Eliashberg
Alessio Corti
Department of Mathematics
Imperial College London
London SW7 2AZ
Mark Haskins
Department of Mathematics
Imperial College London
London SW7 2AZ
Johannes Nordström
Department of Mathematics
Imperial College London
London SW7 2AZ
Tommaso Pacini
Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa