Isolated singularities in $G_2$-structures with torsion

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.

12 pages. v2: exposition expanded, more proofs added. Many typos corrected throughout, and corrected formula appearing in Proposition 13. To appear in Geometriae Dedicata