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Isolated singularities in $G_2$-structures with torsion

Henrique Sá Earp, Jakob Stein

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.

Isolated singularities in $G_2$-structures with torsion

Abstract

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

Paper Structure

This paper contains 9 sections, 20 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Outline and main results
  3. Acknowledgements.
  4. Preliminaries on geometric structures
  5. G2-structures
  6. SU(3)-structures
  7. SU(2)-structures
  8. Circle-invariant G₂-structures
  9. Contact Calabi-Yau Ansatz

Key Result

Theorem 1

Let $M^7= \mathbb{R}_{>0}\times S^3 \times S^3$. There is a one-parameter family $\lbrace(\varphi_\gamma, \psi_\gamma)\rbrace_{\gamma\geq0 }$ of co-closed ${\rm G}_2$-structures on $M$ such that $(\varphi_0, \psi_0)$ is the torsion-free Bryant-Salamon cone and, with respect to the induced metric $g_

Theorems & Definitions (41)

  • Definition 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark
  • Proposition 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Definition 8
  • ...and 31 more