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

Abstract

We develop some consequences of the connection between Calabi–Yau structures and torsion-free G2 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.