Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension

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 ${\mathrm{\Pi }}_{k,\le k{\mu }_k}:={\mathbf{1}}_{(-\mathrm{\infty },k{\mu }_k]}({\mathrm{\square }}_{b,k})$ converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions, where ${\mathrm{\square }}_{b,k}$ denotes the Kohn Laplacian with values in $L^k$. Moreover, when ${\mathrm{\square }}_{b,k}$ possesses a local spectral gap, we show that the Szegő projection ${\mathrm{\Pi }}_k$ 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 ${\mathrm{\Pi }}_{k,({\mathbb{R}}^m\curvearrowright ,\le k\delta )}$ converges to the weighted partial Szegő projection on the high codimension Heisenberg group in the ${\mathcal{C}}^{\mathrm{\infty }}$ 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.