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Deformation theory of G_2 conifolds

Karigiannis, S; Lotay, J; (2020) Deformation theory of G_2 conifolds. Communications in Analysis and Geometry , 28 (5) pp. 1057-1210. 10.4310/CAG.2020.v28.n5.a1. Green open access

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We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $\nu$ 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 $\nu < -4$ 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 $G_2$-manifolds via local rigidity and the cohomogeneity one property of AC $G_2$-manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular $G_2$-cones; and the proof of existence of a "good gauge" needed for desingularization of CS $G_2$-manifolds. Finally, we discuss some open problems.

Type: Article
Title: Deformation theory of G_2 conifolds
Open access status: An open access version is available from UCL Discovery
DOI: 10.4310/CAG.2020.v28.n5.a1
Publisher version: https://doi.org/10.4310/CAG.2020.v28.n5.a1
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: math.DG, math.DG, math.AP, 53C29, 58K60
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1382735
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