AI 中文总结
该研究引入接触符号系统，证明配备接触结构的复射影子流形的接触基本形式构成接触符号系统，且该系统可实现为海森堡对称簇的接触基本形式，还对非辛单李代数的伴随簇证明了接触版本的严格延拓性质。
AI 中文摘要
我们引入了接触符号系统，这是射影基本形式符号系统的非交换类似物，方法是将向量空间上的多项式代数替换为海森堡代数的通用包络代数的分次对偶。对于配备有接触结构的复射影子流形，我们定义了接触基本形式，并证明在一般点处，它们构成一个接触符号系统，这给出了E.嘉当经典结果的接触版本。反过来，我们证明每个接触符号系统都可实现为具有稠密开海森堡轨道的射影簇的接触基本形式，该簇称为与接触符号系统相关联的海森堡对称簇。我们证明，单李代数中射影化幂零轨道的闭包是海森堡对称的当且仅当它是伴随簇，即最小幂零轨道的射影化。对于非辛单李代数的伴随簇，我们利用山口的延拓理论证明了兰兹伯格-马内维尔严格延拓性质的接触类似物。
英文摘要
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.
Comments22 pages
