G-Equivariant Embedding Theorems for CR Manifolds of High Codimension

Abstract

Let $(\mathit{X},{\mathit{T}}^{1,0}\mathit{X})$ be a $(2\mathit{n}+1+\mathit{d})$-dimensional compact CR manifold with codimension $\mathit{d}+1$, $\mathit{d}\ge 1$, and let G be a d-dimensional compact Lie group with CR action on X and T be a globally defined vector field on X such that $\mathbb{C}\mathit{T}\mathit{X}={\mathit{T}}^{1,0}\mathit{X}\oplus {\mathit{T}}^{0,1}\mathit{X}\oplus \mathbb{C}\mathit{T}\oplus \mathbb{C}\underset{\_}{\mathfrak{g}}$, where $\underset{\_}{\mathfrak{g}}$ is the space of vector fields on X induced by the Lie algebra of G. In this work, we show that if X is strongly pseudoconvex in the direction of T and $\mathit{n}\ge 2$, then there exists a G-equivariant CR embedding of X into ${\mathbb{C}}^{\mathit{N}}$ for some $\mathit{N}\in \mathbb{N}$. We also establish a CR orbifold version of Boutet de Monvel’s embedding theorem.