{
  "abstract": "We study Hodge--Riemann relations for ordinary, sequential, and weak $m$-positive real $(1,1)$-forms in complex dimension $d$. For ordinary $m$-positive forms, we prove the fully mixed relations in every permitted bidegree without assuming semipositivity, generalizing results by Xiao. As a generalization of ordinary $m$-positivity, sequential $m$-positivity is defined by the hyperbolicity cones of successive directional derivatives of the determinant. We formulate a generalized mixed Hodge--Riemann conjecture for this condition and prove several cases beyond the scalar inequalities supplied by hyperbolicity. We also consider weak $m$-positivity which corresponds to the pointwise subsolution condition for the complex $m$-Hessian equation. In this setting, we prove the Hodge--Riemann relations for $p\\neq q$ and in several diagonal cases, and show that the associated power map induces a pointwise diffeomorphism onto the cone of positive $(d-1,d-1)$-forms. Using Dinh and Nguyên's $L^2$ globalization method, we pass from the pointwise relations to cohomology classes on compact Kähler manifolds. Finally, we propose conjectures on a numerical criterion for sequential positivity and on solvability of the generalized $m$-Hessian equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03124",
  "author_total": 1,
  "authors": [
    {
      "name": "Enhan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4s96kkevz0bjb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03124",
  "fulltext_url": null,
  "id": "4s96kkevz0bjb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03124",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03124",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4s96kkevz0bjb.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03124",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03124"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Hodge--Riemann relations for $m$-positive and generalized $m$-positive forms",
  "title_text": "Hodge--Riemann relations for m-positive and generalized m-positive forms",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/4s96kkevz0bjb",
  "venue": "arXiv",
  "versions": []
}