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

Spiro Karigiannis

Geometry & Topology 13 (2009) 1583–1655

We present a method to desingularize a compact G2 manifold M with isolated conical singularities by cutting out a neighbourhood of each singular point xi and gluing in an asymptotically conical G2 manifold Ni. 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 M˜ admits a torsion-free G2 structure, with full G2 holonomy.

There are topological obstructions for this procedure to work, which arise from the degree 3 and degree 4 cohomology of the asymptotically conical G2 manifolds Ni 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 G2 structure on an asymptotically conical G2 manifold N under an appropriate gauge-fixing condition, which is required to make the gluing procedure work.

$\mathrm{G}_2$ manifolds, conical singularity, desingularization, asymptotically conical manifold
Mathematical Subject Classification 2000
Primary: 53C29
Secondary: 58J05
Received: 23 July 2008
Revised: 23 October 2008
Accepted: 10 February 2009
Published: 3 March 2009
Proposed: Simon Donaldson
Seconded: Ron Fintushel and Ron Stern
Spiro Karigiannis
Mathematical Institute
University of Oxford
24-29 St Giles’
Oxford OX1 3LB
United Kingdom
Department of Pure Mathematics
University of Waterloo
200 University Avenue West
Waterloo, Ontario N2L 3G1