Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds

Abstract

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n⩾2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in \{0,1,\dots ,n-1\}$, let ${\square }_{b,k}^{(q)}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda \ge 0$, let ${\Pi }_{k,\le \lambda }^{(q)}:=E((-\infty ,\lambda ])$, where $E$ denotes the spectral measure of ${\square }_{b,k}^{(q)}$. In this work, we prove that ${\Pi }_{k,\le k^{-N_0}}^{(q)}{F}_k^{\ast }$, $F_k{\Pi }_{k,\le k^{-N_0}}^{(q)}{F}_k^{\ast }$, $N_0\ge 1$, admit asymptotic expansions on the non-degenerate part of the characteristic manifold of ${\square }_{b,k}^{(q)}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k{\Pi }_{k,\le 0}^{(q)}{F}_k^{\ast }$ admits a full asymptotic expansion if ${\square }_{b,k}^{(q)}$ has small spectral gap property with respect to $F_k$ and ${\Pi }_{k,\le 0}^{(q)}$ is $k$-negligible away the diagonal with respect to $F_k$. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ actions.

arXiv comment106 pages, to appear in the Memoirs of the American Mathematical Society