Bryant-Salamon $\mathrm{G}_2$ manifolds and coassociative fibrations

Abstract

Bryant-Salamon constructed three 1-parameter families of complete manifolds with holonomy $\mathrm{G}_2$ which are asymptotically conical to a holonomy $\mathrm{G}_2$ cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on $\mathbb R^7$: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of $\mathbb C^3$ with a line, and the Harvey-Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere $\mathcal{S}^4$, and the cone on $\mathbb C \mathbb P^3$, whose smooth fibres are $T^*\mathcal{S}^2$, and whose singular fibres are $\mathbb R^4/\{\pm 1\}$. We relate these fibrations to hypersymplectic geometry, Donaldson's work on Kovalev-Lefschetz fibrations, harmonic 1-forms and the Joyce--Karigiannis construction of holonomy $\mathrm{G}_2$ manifolds, and we construct vanishing cycles and associative "thimbles" for these fibrations.