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Desingularization of Coassociative 4folds with Conical Singularities
, 2006
"... Given a coassociative 4fold N with a conical singularity in a ϕclosed 7manifold M (a manifold endowed with a distinguished closed 3form ϕ), we construct a smooth family, {N ′ (t) : t ∈ (0, τ)} for some τ> 0, of (smooth, nonsingular,) compact coassociative 4folds in M which converge to N in t ..."
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Cited by 6 (3 self)
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Given a coassociative 4fold N with a conical singularity in a ϕclosed 7manifold M (a manifold endowed with a distinguished closed 3form ϕ), we construct a smooth family, {N ′ (t) : t ∈ (0, τ)} for some τ> 0, of (smooth, nonsingular,) compact coassociative 4folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4fold in R 7, dilated by t, then deforming the resulting compact submanifold of M to the required coassociative 4fold. 1
Special Lagrangian conifolds. II: Gluing constructions
 in Cm., Proc. Lond. Math. Soc
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Deformation theory of G2 conifolds
, 2014
"... We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G2 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, − 5 2), then the moduli space is smooth and we ..."
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We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G2 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, − 5 2), then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates ν < −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 a finitecodimensional subspace of the moduli space, which is often the full moduli space. We also present many applications of these results, including: the local rigidity of the Bryant– Salamon AC G2 manifolds; an extension of our deformation theory in the AC case to higher rates under certain natural assumptions; the cohomogeneity one property of AC G2 manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular G2 cones; and the proof of existence of a “good gauge ” needed for desingularization of CS G2 manifolds. Finally, we discuss some open problems for future study.