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

Abstract

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 $M/⟨\iota ⟩$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form ${\mathbb{R}}^3\times ({\mathbb{R}}^4/\{\pm 1\})$. 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.

83 pages. Version 3: Fixed three grammar mistakes and added a missing parenthesis. Final version to appear in Journal of Differential Geometry

References in corpus (1)

• On the structure of supersymmetric T3 fibrations

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