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

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

doi:10.2140/gt.2013.17.1955

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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

DOI: 10.2140/gt.2013.17.1955

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 $ℙ^{1}$ s with normal bundle $O (- 1) \oplus O (- 1)$ , 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 $G_{2}$ –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.

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

Mathematical Subject Classification 2000

Primary: 14J30, 53C29

Secondary: 14E15, 14J28, 14J32, 14J45, 53C25

References

Publication

Received: 24 August 2012

Revised: 2 March 2013

Accepted: 4 March 2013

Published: 15 July 2013

Proposed: Richard Thomas

Seconded: Ronald Fintushel, Yasha Eliashberg

Authors

Alessio Corti
Department of Mathematics Imperial College London London SW7 2AZ UK

Mark Haskins
Department of Mathematics Imperial College London London SW7 2AZ UK

Johannes Nordström
Department of Mathematics Imperial College London London SW7 2AZ UK

Tommaso Pacini
Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa Italy

Volume 17, issue 4 (2013)