Deformation theory of $\mathrm{G}_2$ conifolds

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $\mathrm{G}_2$ manifolds. In the AC case, we show that if the rate of convergence ν to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates $\nu \lt -4$ in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space.

We also present many applications of these results, including: the uniqueness of the Bryant–Salamon AC $\mathrm{G}_2$ manifolds via local rigidity and the cohomogeneity one property of AC $\mathrm{G}_2$ manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular $\mathrm{G}_2$ cones; and the proof of existence of a “good gauge” needed for desingularization of CS $\mathrm{G}_2$ manifolds. Finally, we discuss some open problems.

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A very small portion of this research was completed while the first author was supported by a Marie Curie Fellowship of the European Commission under contract number MIF1-CT-2006-039113.

The second author was supported by an EPSRC Career Acceleration Fellowship EP/H003584/1 and EPSRC grant EP/K010980/1.

The contents of this work reflect only the authors’ views and not the views of the European Commission.

Received 24 January 2013

Accepted 20 December 2017

Published 14 October 2020