A ddΦ-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds
Abstract
We study the properties of the ddΦ-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures Φ. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the ddΦ-operator and obtain an analogue of the ∂ ∂̅-lemma in Kähler geometry. Using this, we define Bott--Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of Kähler identities and Bryant--Harvey's identities in the G2-case (R. Bryant, Some remarks on G2-structures, Proceedings of the 11th and 12th Gökova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.
Related
- Integral Bott--Chern cohomology of Generalized Calabi--Eckmann threefolds
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BibTeX
@misc{dwivedi2026ddφlemma,
title = {{A $dd^Φ$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds}},
author = {Shubham Dwivedi and Ragini Singhal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04156},
eprint = {2609.04156},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04156},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A dd^Φ-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds
AU - Shubham Dwivedi
AU - Ragini Singhal
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04156
UR - https://arxiv.org/abs/2609.04156
AB - We study the properties of the dd^Φ-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures Φ. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the dd^Φ-operator and obtain an analogue of the ∂ ∂̅-lemma in Kähler geometry. Using this, we define Bott--Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of Kähler identities and Bryant--Harvey's identities in the G_2-case (R. Bryant, Some remarks on G_2-structures, Proceedings of the 11th and 12th Gökova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.
ER -