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

作者：Corti Alessio^*; Haskins Mark; Nordstroem Johannes; Pacini Tommaso

来源：Geometry and Topology, 2013, 17(4): 1955-2059.

DOI：10.2140/gt.2013.17.1955

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) circle plus 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 [17] 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.

• 出版日期2013