Deformation theory of G_2 conifolds

Abstract

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.

106 pages, no figures. Version 5: an error was found in Lemma 4.54 when checking the final proofs for publication. The statement and proof of this lemma have now been corrected, and no other results were affected. Also two references were updated. Final version to appear in CAG

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• New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres
• Asymptotically conical Calabi-Yau manifolds, I
• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Deformations of asymptotically cylindrical G_2 manifolds

Cited by in corpus (10)

• Instantons on the exceptional holonomy manifolds of Bryant and Salamon
• Monopoles on the Bryant-Salamon $G_2$ Manifolds
• Deformation theory of nearly Kähler manifolds
• Infinitesimal moduli of G2 holonomy manifolds with instanton bundles
• Bryant-Salamon ${\mathrm{G}}_2$ manifolds and coassociative fibrations
• Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds
• Deformations of Asymptotically Conical $G_2$-Instantons
• Deformation Theory of Asymptotically Conical Spin(7)-Instantons
• Singular foliations for M-theory compactification
• $G_2$-instantons on the ALC members of the ${\mathbb{B}}_7$ family