#### Volume 13, issue 3 (2009)

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

### Spiro Karigiannis

Geometry & Topology 13 (2009) 1583–1655
##### 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 $\stackrel{˜}{M}$ 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.

##### Keywords
$\mathrm{G}_2$ manifolds, conical singularity, desingularization, asymptotically conical manifold
Primary: 53C29
Secondary: 58J05