G$_2$ manifolds with nodal singularities along circles

TL;DR

This paper constructs compact G$_2$ manifolds with nodal singularities along circles using a twisted connected sum approach, addressing complex obstructions and proposing potential resolutions for singularities.

Contribution

It introduces a method to build G$_2$ manifolds with circle singularities via twisted connected sums and analyzes the associated infinite-dimensional obstructions.

Findings

Matching building blocks found by solving Calabi conjecture.

Obstruction space for the construction is infinite dimensional.

Evidence suggests further gluing could resolve obstructions and produce smooth G$_2$ manifolds.

Abstract

The goal of this paper is the construction of a compact manifold with G$_2$ 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 G$_2$ holonomy and isolated conical singularities with link $\mathbb{S}^3\times\mathbb{S}^3$.