Desingularization of Coassociative 4-folds with Conical Singularities: Obstructions and Applications

TL;DR

This paper investigates the process of desingularizing coassociative 4-folds with conical singularities through gluing, addressing obstructions, and exploring implications for moduli spaces and applications in G_2 manifolds.

Contribution

It extends previous work by analyzing obstructions to desingularization, linking them to geometric stability, and relating moduli spaces of singular and smooth coassociative 4-folds.

Findings

Identified geometric obstructions to desingularization.

Established relationships between moduli spaces of singular and smooth coassociative 4-folds.

Applied theory to examples in compact G_2 manifolds.

Abstract

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work 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 G_2 manifolds.