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

TL;DR

This paper proves the existence of a vast new class of asymptotically cylindrical Calabi-Yau 3-folds derived from weak Fano 3-folds, significantly expanding known examples and enabling applications in G_2-manifold construction.

Contribution

It establishes a general method to construct and analyze ACyl Calabi-Yau 3-folds from weak Fano 3-folds, especially semi-Fano types, and computes their topological invariants.

Findings

Hundreds of thousands of new ACyl Calabi-Yau 3-folds identified.

Semi-Fano 3-folds have unique topological properties and contain special rational curves.

Methods developed for topological invariant computation are broadly applicable.

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^1s with normal bundle O(-1) + 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.