Gluing and deformation of asymptotically cylindrical Calabi-Yau manifolds in complex dimension three

TL;DR

This paper advances the understanding of Calabi-Yau threefolds by improving gluing techniques, proving smoothness of their moduli space, and establishing the local diffeomorphism property of the gluing map in the asymptotically cylindrical setting.

Contribution

It provides new proofs for gluing Calabi-Yau threefolds, extends moduli space smoothness results to asymptotically cylindrical cases, and demonstrates the local diffeomorphism nature of the gluing map.

Findings

Enhanced gluing method for asymptotically cylindrical Calabi-Yau threefolds

Proved smoothness of the Calabi-Yau moduli space in the asymptotically cylindrical case

Established the local diffeomorphism property of the gluing map

Abstract

We develop some consequences of the connection between Calabi-Yau structures and torsion-free $G_2$ structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi-Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi-Yau structures on a six-dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi-Yau threefolds, extended between these moduli spaces, is a local diffeomorphism: that is, that every deformation of a glued Calabi-Yau threefold arises from an essentially unique deformation of the asymptotically cylindrical pieces.