Year: 2026
Type: preprint
Abstract: In this paper, we consider a CR manifold $X$ of high codimension and a CR line bundle $L$ over $X$. For each $k \in \mathbb{N}$, let $L^k$ denote the $k$-th power of $L$. We establish that the spectral projection $Π_{k,\leq kμ_k} := \mathbf{1}_{(-\infty, kμ_k]}(\Box_{b,k})$ converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions, where $\Box_{b,k}$ denotes the Kohn Laplacian with values in $L^k$. Moreover, when $\Box_{b,k}$ possesses a local spectral gap, we show that the Szegő projection $Π_k$ converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions. We also provide an explicit formula for the Szegő projection on the high codimension Heisenberg group. Furthermore, we consider a high codimension CR manifold $X$ equipped with a transversal and CR $\mathbb{R}^m$-action. In this setting, we prove that the partial Szegő projection $Π_{k,({\mathbb R}^m\curvearrowright,\leq kδ)}$ converges to the weighted partial Szegő projection on the high codimension Heisenberg group in the $\mathcal{C}^\infty$ topology. As an application of our results, we establish a Kodaira embedding theorem for high codimension CR manifolds with a transversal and CR $\mathbb{R}^m$-action.
Source: arXiv (Cornell University)
Authors: Hsiao, Chin-Yu, Yang, Chin Jui
Institution: National Taiwan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223061351