Desingularization of $\mathrm{G}_2$ manifolds with isolated conical singularities

Abstract

We present a method to desingularize a compact $G_2$ manifold $M$ with isolated conical singularities by cutting out a neighbourhood of each singular point $x_i$ and gluing in an asymptotically conical $G_2$ manifold $N_i$. Controlling the error on the overlap gluing region enables us to use a result of Joyce to conclude that the resulting compact smooth $7$–manifold $\tilde{M}$ admits a torsion-free $G_2$ structure, with full $G_2$ holonomy.

There are topological obstructions for this procedure to work, which arise from the degree $3$ and degree $4$ cohomology of the asymptotically conical $G_2$ manifolds $N_i$ which are glued in at each conical singularity. When a certain necessary topological condition on the manifold $M$ with isolated conical singularities is satisfied, we can introduce correction terms to the gluing procedure to ensure that it still works. In the case of degree $4$ obstructions, these correction terms are trivial to construct, but in the case of degree $3$ obstructions we need to solve an elliptic equation on a noncompact manifold. For this we use the Lockhart–McOwen theory of weighted Sobolev spaces on manifolds with ends. This theory is also used to obtain a good asymptotic expansion of the $G_2$ structure on an asymptotically conical $G_2$ manifold $N$ under an appropriate gauge-fixing condition, which is required to make the gluing procedure work.