Contact fundamental forms and adjoint varieties

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.

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BibTeX RIS

BibTeX

@misc{fu2026contact,
  title = {{Contact fundamental forms and adjoint varieties}},
  author = {Baohua Fu and Jun-Muk Hwang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02105},
  eprint = {2609.02105},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02105},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Contact fundamental forms and adjoint varieties
AU  - Baohua Fu
AU  - Jun-Muk Hwang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02105
UR  - https://arxiv.org/abs/2609.02105
AB  - 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.
ER  -

Abstract

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