$$\hbox {G}_2$$ Manifolds with Nodal Singularities Along Circles

Abstract

The goal of this paper is the construction of a compact manifold with G2 holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G2 holonomy and isolated conical singularities with link S 3 × S 3 .

Atiyah classes and the essential obstructions in deforming a singular $G_{2}-$instanton

“…F 0 A = 0 at p. Because v is non-zero on a dense open set on S 5 , F 0 A = 0 on the same set. By continuity of F 0 A , it vanishes everywhere on S 5 . When E is a twisted tangent bundle of P 2 , by Lemma 4.5, the injection ⊟ is an isomorphism since the dimension of the domain equals the dimension of the range.…”

“…Gram-Schmit process for each eigen-space of P yields a complete orthonormal P −eigenbasis (φ β , β ∈ Spec mul P ) for L 2 [S 5 , Dom] such that • the eigen-section ζ perpendicular to the Atiyah classes in condition III appears as an element in the basis if essential obstruction is non-trivial,…”

“…Finite dimensional obstructions can prevent a gluing construction: see the work of Brendle Kapouleas [4] on Einstein metrics. Infinite dimensional obstructions make it even harder: see the work of Chen [5] on twisted connected sum of G 2 −structures with conical singularities along circles. An option is to add parameters into the domain Banach space.…”