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 ${\mathbb{P}}^1$s with normal bundle $\mathcal{O}\left(-1\right)\oplus \mathcal{O}\left(-1\right)$, 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.