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

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 a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the $G_2$-structure. Then $M / {\langle \iota \rangle}$ is a $G_2$- orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb{R}^3 \times (\mathbb{R}^4 / {\lbrace \pm 1 \rbrace})$. We resolve this orbifold by gluing in a family of Eguchi–Hanson spaces, parametrized by a nonvanishing 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 the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi–Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.

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