Desingularization of G2 manifolds with isolated conical singularities

Karigiannis, Spiro

doi:10.2140/gt.2009.13.1583

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

Spiro Karigiannis

Geometry & Topology 13 (2009) 1583–1655

DOI: 10.2140/gt.2009.13.1583

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.

Keywords

$\mathrm{G}_2$ manifolds, conical singularity, desingularization, asymptotically conical manifold

Mathematical Subject Classification 2000

Primary: 53C29

Secondary: 58J05

References

Publication

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

Authors

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 Canada
http://www.math.uwaterloo.ca/~karigiannis

Volume 13, issue 3 (2009)