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February 2021 A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces
Dominic Joyce, Spiro Karigiannis
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J. Differential Geom. 117(2): 255-343 (February 2021). DOI: 10.4310/jdg/1612975017

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 $M / {\langle \iota \rangle}$ 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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Dominic Joyce. Spiro Karigiannis. "A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces." J. Differential Geom. 117 (2) 255 - 343, February 2021. https://doi.org/10.4310/jdg/1612975017

Information

Received: 28 July 2017; Published: February 2021
First available in Project Euclid: 10 February 2021

Digital Object Identifier: 10.4310/jdg/1612975017

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 2 • February 2021
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