Abstract
We present a method to desingularize a compact manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and gluing in an asymptotically conical manifold . Controlling the error on the overlap gluing region enables us to use a result of Joyce to conclude that the resulting compact smooth –manifold admits a torsion-free structure, with full holonomy.
There are topological obstructions for this procedure to work, which arise from the degree and degree cohomology of the asymptotically conical 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 gluing procedure to ensure that it still works. In the case of degree obstructions, these correction terms are trivial to construct, but in the case of degree 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 structure on an asymptotically conical manifold under an appropriate gauge-fixing condition, which is required to make the gluing procedure work.
Citation
Spiro Karigiannis. "Desingularization of $\mathrm{G}_2$ manifolds with isolated conical singularities." Geom. Topol. 13 (3) 1583 - 1655, 2009. https://doi.org/10.2140/gt.2009.13.1583
Information