Conically singular Cayley submanifolds III: Fibrations

TL;DR

This paper establishes stability results for Cayley fibrations with conical singularities, including Morse type, and constructs new examples on twisted connected sum G2 manifolds, advancing the understanding of Cayley fibrations.

Contribution

It provides the first stability results for Cayley fibrations with conical singularities and constructs new examples on twisted connected sum G2 manifolds.

Findings

Stability of weak Cayley fibrations with minimal assumptions

Stability of Cayley fibrations under stronger geometric conditions

Construction of Cayley and coassociative fibrations on twisted connected sum G2 manifolds

Abstract

This is the third and last in a series of papers working towards the construction of non-trivial Cayley fibrations using gluing methods. In this paper we will show two stability results for Cayley fibrations with certains types of conical singularities (in particular Morse type singularities present in holomorphic fibrations of Calabi--Yau fourfolds). The first is a stability result for weak fibrations, which has minimal assumptions. Then we show stability of Cayley fibrations in the usual sense. This requires stronger geometric assumptions on the Cayley cone and the initial fibration. As an application we construct examples of Cayley fibrations on twisted connected sum $G_2$ manifolds times a circle. In particular we also obtain examples of coassociative fibrations of twisted connected sum $G_2$ manifolds, completing the longstanding programme by Kovalev.