Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds - Corti,A; Haskins,M; Nordström,J; Pacini,T (Geometry & Topology, 2013)
We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–foldsstarting with (almost) any deformation family of smooth weak Fano 3–folds. Thisallow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previouslyonly a few hundred ACyl Calabi–Yau 3–folds were known. We pay particularattention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano3–folds satisfy stronger cohomology vanishing theorems and enjoy certaintopological properties not satisfied by general weak Fano 3–folds, but are far morenumerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P1s withnormal bundle O.1/ O.1/,giving rise to compact rigid holomorphic curves inthe associated ACyl Calabi–Yau 3–folds.We introduce some general methods to compute the basic topological invariants ofACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a smallnumber of representative examples in detail. Similar methods allow the computationof the topology in many other examples.All the features of the ACyl Calabi–Yau 3–folds studied here find application in [17]where we construct many new compact G2 –manifolds using Kovalev’s twistedconnected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano3–folds are particularly well-adapted for this purpose.