A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces

摘要

We give a new construction of compact Riemannian 7-manifolds with holonomy$G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy aproper subgroup of $G_2$) such that $M$ admits an involution $iota$ preservingthe $G_2$-structure. Then $M/{langle iota angle}$ is a $G_2$-orbifold, withsingular set $L$ an associative submanifold of $M$, where the singularities arelocally of the form $mathbb R^3 imes (mathbb R^4 / {pm 1})$. We resolvethis orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by anonvanishing closed and coclosed $1$-form $lambda$ on $L$. Much of the analytic difficulty lies in constructing appropriate closed$G_2$-structures with sufficiently small torsion to be able to apply thegeneral existence theorem of the first author. In particular, the constructioninvolves solving a family of elliptic equations on the noncompact Eguchi-Hansonspace, parametrized by the singular set $L$. We also present twogeneralizations of the main theorem, and we discuss several methods ofproducing examples from this construction.