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)⊕ O (− 1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi–Yau 3–folds.
Mathematical Sciences Publishers
15 Jul 2013
Volume: 17 Issue: 4 Pages: 1955-2059
Geometry & Topology