Abstract
We prove that the moduli space of holonomy G2-metrics on a closed 7-manifold can be disconnected by presenting a number of explicit examples. We detect different connected components of the G2-moduli space by defining an analytic refinement ν¯(M,g)∈Z of the defect invariant ν(M,φ)∈Z/48 of G2-structures φ on a closed 7-manifold M introduced by the first and third authors. The ν¯-invariant is defined using η-invariants and Mathai-Quillen currents on M and we compute it for twisted connected sums à la Kovalev, Corti-Haskins-Nordström-Pacini and extra-twisted connected sums as constructed by the second and third authors. In particular, we find examples of G2-holonomy metrics in different components of the moduli space where the associated G2-structures are homotopic and other examples where they are not.
Original language | English |
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Pages (from-to) | 865-907 |
Number of pages | 43 |
Journal | Inventiones Mathematicae |
Volume | 239 |
Issue number | 3 |
Early online date | 23 Jan 2025 |
DOIs | |
Publication status | Published - 31 Mar 2025 |
Acknowledgements
The authors thank Uli Bunke, Alessio Corti, Mark Haskins, Matthias Lesch, Arkadi Schelling and Thomas Walpuski for valuable discussions, and the referees for constructive comments. SG and JN would like to thank the Simons foundation for its support of their research under the Simons Collaboration on “Special Holonomy in Geometry, Analysis and Physics” (grants #488617, Sebastian Goette, and #488631, Johannes Nordström).Funding
The authors thank Uli Bunke, Alessio Corti, Mark Haskins, Matthias Lesch, Arkadi Schelling and Thomas Walpuski for valuable discussions, and the referees for constructive comments. SG and JN would like to thank the Simons foundation for its support of their research under the Simons Collaboration on \u201CSpecial Holonomy in Geometry, Analysis and Physics\u201D (grants #488617, Sebastian Goette, and #488631, Johannes Nordstr\u00F6m).
Funders | Funder number |
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Simons Foundation | 488617, 488631 |
Keywords
- math.GT
- math.DG
- 57R20 (Primary) 53C29, 58J28 (Secondary)