Isolated singularities in $G_2$-structures with torsion

TL;DR

This paper investigates $G_2$-structures with torsion and isolated singularities, focusing on symmetries and the limitations of collapsing circle fibers in contact Calabi-Yau manifolds to produce such structures.

Contribution

It characterizes circle-invariant $G_2$-structures with singularities and demonstrates the impossibility of creating bounded torsion $G_2$-structures through collapsing circle fibers.

Findings

Many conical singularity examples admit additional symmetries

Circle-invariant $G_2$-structures are described in symmetric contexts

Collapsing circle fibers cannot produce bounded torsion $G_2$-structures

Abstract

We revisit the study of $G_2$-structures with special torsion, and isolated singularities. Many of the known examples with conical singularities admit additional symmetries, and we describe circle-invariant $G_2$-structures in this context. Finally, we show that collapsing the circle fibres of a contact Calabi-Yau manifold at isolated points cannot produce a $G_2$-structure with bounded torsion.