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 ${ℂ}^3$ 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 $ℂ{\mathbb{P}}^3$, whose smooth fibres are $T^{\ast }{\mathcal{S}}^2$, 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.

布莱恩特-萨拉蒙构建了三个完整的具有完整性的1参数系列流形 ${\mathrm{G}}_2$ 渐近于完整的圆锥形 ${\mathrm{G}}_2$锥体。对于这些家庭中的每个家庭，包括他们的渐近锥，我们通过渐近圆锥形和圆锥形奇异的四联体构造纤维化。我们表明，这些纤维化是以下三种众所周知的共缔合纤维化的自然概括。${\mathbb{[R}}^7$：4平面的琐碎纤维化，是标准Lefschetz纤维化的产物 ${ℂ}^3$一条线，以及Harvey-Lawson缔合纤维化。特别是，我们描述了在4球面上的反自对偶2形式束的缔合纤维化${\mathcal{小号}}^4$和锥形 $ℂ{\mathbb{P}}^3$，其光滑纤维是 ${Ť}^{\ast }{\mathcal{小号}}^2$，并且其奇异纤维是 ${\mathbb{[R}}^4∕\{\pm \mathrm{1个}\}$。我们将这些纤维与高辛几何，唐纳森关于Kovalev–Lefschetz纤维，调和1形式以及乔伊斯·卡里吉安尼斯的完整学的工作联系起来${\mathrm{G}}_2$ 歧管，我们为这些纤维化建立消失周期和相关的“顶针”。