{
  "abstract": "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 $\\partial \\bar{\\partial}$-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 $\\mathrm{G}_2$-case (R. Bryant, Some remarks on $\\mathrm{G}_2$-structures, Proceedings of the 11th and 12th Gökova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04156",
  "author_total": 2,
  "authors": [
    {
      "name": "Shubham Dwivedi"
    },
    {
      "name": "Ragini Singhal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kvb5rtj9xxjy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04156",
  "fulltext_url": null,
  "id": "4kvb5rtj9xxjy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04156",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7208777324",
  "pdf_url": "https://arxiv.org/pdf/2609.04156",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kvb5rtj9xxjy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.04156",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04156"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A $dd^Φ$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds",
  "title_text": "A dd^Φ-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/4kvb5rtj9xxjy",
  "venue": "arXiv",
  "versions": []
}