Szego Kernel Asymptotics for High Power of Cr Line Bundles and Kodaira Embedding Theorems on Cr Manifolds (Memoirs of the American Mathematical Society)

By Chin-Yu Hsiao

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant 2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in \{0,1,\ldots,n-1\}$, let $\Box ^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda \geq 0$, let $\Pi ^{(q)}_{k,\leq \lambda} :=E((-\infty,\lambda ])$, where $E$ denotes the spectral measure of $\Box ^{(q)}_{b,k}$. In this work, the author proves that $\Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq 1$, 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 $F_k\Pi ^{(q)}_{k,\leq 0}F^*_k$ 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 $\Pi^{(q)}_{k,\leq 0}$ is $k$-negligible away the diagonal with respect to $F_k$. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ action.

• Format: Paperback
• Language: English
• ISBN: 1470441012
• ISBN13: 9781470441012
• Release Date: June 2018
• Publisher: Amer Mathematical Society
• Length: 140 Pages
• Weight: 0.54 lbs.
• Dimensions: 10.3" x 0.5" x 7.3"