Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Fu, Baohua, Hwang, Jun-Muk
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207729725
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