Coassociative 4-folds with conical singularities

Lotay, Jason D.

doi:10.4310/cag.2007.v15.n5.a1

Cited by 19 publications

(56 citation statements)

References 13 publications

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“…Remark 5.14. Use the notation in [16]. By Lemma 5.12, M C,D,E is a coassociative submanifold with conical singularities when (i) C > 0, D > 0, E = 0,…”

Section: So(3) = So(3) × {I 2 }-Actionmentioning

confidence: 99%

Cohomogeneity one coassociative submanifolds in the bundle of anti-self-dual 2-forms over the 4-sphere

Kawai

2018

Communications in Analysis and Geometry

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Coassociative submanifolds are 4-dimensional calibrated submanifolds in G2-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in Λ 2 − S 4 , which is the complete G2-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3-or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of Λ 2 − S 4 up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical singularities and their desingularizations.We give a summary about real irreducible representations in [5,18].Definition A.1. Let G be a compact Lie group and (V, ρ) be a C-irreducible representation of G. We call (V, ρ) self-conjugate if V has a conjugate linear map J on V satisfyingThis map is called a structure map. A self-conjugate representation (V, ρ) is said to be of index ±1 if J 2 = ±1.Proposition A.2. Let (V, ρ) be a C-irreducible representation of G. Then one of the following is satisfied.1. (V, ρ) is a self-conjugate representation of index 1. In this case, (V, ρ) is a complexification of a real representation.   ,   J J 0   for (B.2),

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“…Remark 5.14. Use the notation in [16]. By Lemma 5.12, M C,D,E is a coassociative submanifold with conical singularities when (i) C > 0, D > 0, E = 0,…”

Section: So(3) = So(3) × {I 2 }-Actionmentioning

confidence: 99%

Cohomogeneity one coassociative submanifolds in the bundle of anti-self-dual 2-forms over the 4-sphere

Kawai

2018

Communications in Analysis and Geometry

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“…The following definition is independent of choice of equivalent Spin(7)coordinate system. It is analogous to [17,Defn 3.5]. On a Calabi-Yau manifold M we are given a Ricci-flat metric ω that we often have no explicit expression for.…”

Section: Deformations Of Conically Singular Cayley Submanifoldsmentioning

confidence: 99%

Deformations of Conically Singular Cayley Submanifolds

Moore

2018

J Geom Anal

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In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a Calabi-Yau four-fold we show by comparing Cayley and complex deformations that in this special case the moduli space is a smooth manifold. We also perform calculations of some of the quantities discussed for some examples.

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“…The author extended this result in [16] and [18] to the CS and AC settings, where various similarities and differences occur which shall be discussed later. These results will be crucial in understanding obstructions to the gluing problem.…”

Section: Definition 211mentioning

confidence: 90%

“…(a) By [16,Proposition 3.6], if N is a CS coassociative 4-fold then the cones at the singularities are coassociative in R 7 .…”

Section: Remarksmentioning

confidence: 99%

Desingularization of coassociative 4-folds with conical singularities: Obstructions and applications

Lotay¹

2014

Trans. Amer. Math. Soc.

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We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends work in [17] on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for deformations of conically singular coassociative 4-folds, and thus relate them to the stability of the singularities. We use our results to describe the relationship between moduli spaces of coassociative 4-folds with conical singularities and those of their desingularizations. We also apply our theory in examples, including to the known conically singular coassociative 4-folds in compact holonomy G2 manifolds.

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Coassociative 4-folds with conical singularities

Cited by 19 publications

References 13 publications

Cohomogeneity one coassociative submanifolds in the bundle of anti-self-dual 2-forms over the 4-sphere

Cohomogeneity one coassociative submanifolds in the bundle of anti-self-dual 2-forms over the 4-sphere

Deformations of Conically Singular Cayley Submanifolds

Desingularization of coassociative 4-folds with conical singularities: Obstructions and applications

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