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- W2884257857 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 $mathbb R^3 times (mathbb R^4 / {pm 1})$. 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." @default.
- W2884257857 created "2018-08-03" @default.
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- W2884257857 date "2021-02-01" @default.
- W2884257857 modified "2023-10-01" @default.
- W2884257857 title "A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces" @default.
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- W2884257857 doi "https://doi.org/10.4310/jdg/1612975017" @default.
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