Szeg\"{o} kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds

Chin-Yu Hsiao Institute of Mathematics, Academia Sinica, Taiwan

Complex Variables and Complex Analysis mathscidoc:1803.08008

Memoirs of the American Mathematical Society, 254, (1217), 1--146pp, 2018.6
Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda\geq0$, let $\Pi^{(q)}_{k,\leq\lambda}:=E((-\infty,\lambda])$, where $E$ denotes the spectral measure of $\Box^{(q)}_{b,k}$. In this work, we prove that $\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq1$, admit asymptotic expansions with respect to $k$ on the non-degenerate part of the characteristic manifold of $\Box^{(q)}_{b,k}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k\Pi^{(q)}_{k,\leq0}F^*_k$ admits a full asymptotic expansion with respect to $k$ if $\Box^{(q)}_{b,k}$ has small spectral gap property with respect to $F_k$ and $\Pi^{(q)}_{k,\leq0}$ 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 $S^1$ action.
Szego kernels, Complex Fourier integral operators, CR Kodaira embedding theorem, CR geometry
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  • See the link https://bookstore.ams.org/memo-254-1217/ or http://www.ams.org/books/memo/1217/
@inproceedings{chin-yu2018szeg\"{o},
  title={Szeg\"{o} kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds},
  author={Chin-Yu Hsiao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180330000017868691001},
  booktitle={Memoirs of the American Mathematical Society},
  volume={254},
  number={1217},
  pages={1--146pp},
  year={2018},
}
Chin-Yu Hsiao. Szeg\"{o} kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds. 2018. Vol. 254. In Memoirs of the American Mathematical Society. pp.1--146pp. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180330000017868691001.
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