Bryant–Salamon G2 manifolds and coassociative fibrations - Karigiannis, S; Lotay, JD (Elsevier, 2020)
Bryant–Salamon constructed three 1-parameter families of complete manifolds with holonomy G2 which are asymptotically conical to a holonomy G2 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 R7: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of C3 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 y4, and the cone on CP3, whose smooth fibres are T* y2, and whose singular fibres are R4 / {+-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 G2 manifolds, and we construct vanishing cycles and associative “thimbles” for these fibrations.