Desingularization of G_2 manifolds with isolated conical singularities

TL;DR

This paper develops a method to smooth out isolated conical singularities in compact G_2 manifolds by gluing in asymptotically conical G_2 manifolds, ensuring the existence of torsion-free G_2 structures under certain topological conditions.

Contribution

It introduces a novel desingularization technique for G_2 manifolds with conical singularities, incorporating correction terms and elliptic equation solutions to handle topological obstructions.

Findings

Successfully desingularized G_2 manifolds with conical singularities.

Established conditions under which the gluing procedure yields smooth G_2 structures.

Applied weighted Sobolev space theory to control asymptotic behavior and solve elliptic equations.

Abstract

We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the resulting compact smooth 7-manifold 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 which are glued in at each conical singularity. When a certain necessary topological condition on the manifold with isolated conical singularities is satisfied, we can introduce correction terms to the glueing 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 non-compact 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 under an appropriate gauge-fixing condition, which is required to make the glueing procedure work.