{
  "abstract": "We introduce contact symbol systems, a noncommutative analogue of symbol systems for projective fundamental forms, by replacing the polynomial algebra on a vector space by the graded dual of the universal enveloping algebra of a Heisenberg algebra. For a complex projective submanifold equipped with a contact structure, we define contact fundamental forms and prove that, at a general point, they form a contact symbol system, which gives a contact version of the classical result due to E. Cartan. Conversely, we prove that every contact symbol system can be realized as the contact fundamental forms of a projective variety with a dense open Heisenberg orbit, called the Heisenberg-symmetric variety associated to the contact symbol system. We show that the closure of a projectivized nilpotent orbit in a simple Lie algebra is Heisenberg-symmetric if and only if it is the adjoint variety, namely, the projectivization of the minimal nilpotent orbit. For adjoint varieties of non-symplectic simple Lie algebras, we prove the contact analogue of the Landsberg--Manivel strict prolongation property by using Yamaguchi's prolongation theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02105",
  "author_total": 2,
  "authors": [
    {
      "name": "Baohua Fu"
    },
    {
      "name": "Jun-Muk Hwang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db41w15e5c5g9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02105",
  "fulltext_url": null,
  "id": "db41w15e5c5g9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02105",
  "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.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7207729725",
  "pdf_url": "https://arxiv.org/pdf/2609.02105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db41w15e5c5g9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Contact fundamental forms and adjoint varieties",
  "title_text": "Contact fundamental forms and adjoint varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/db41w15e5c5g9",
  "venue": "arXiv",
  "versions": []
}