Bryant–Salamon G2 manifolds and coassociative fibrations

Abstract Bryant–Salamon constructed three 1-parameter families of complete manifolds with holonomy G 2 which are asymptotically conical to a holonomy 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 R 7 : the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of 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 S 4 , and the cone on ℂ P 3 , whose smooth fibres are T ∗ S 2 , and whose singular fibres are R 4 ∕ { ± 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 G 2 manifolds, and we construct vanishing cycles and associative “thimbles” for these fibrations.