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

About this Title

Chin-Yu Hsiao, Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No.1, Sec.4, Roosevelt Road, Taipei 10617, Taiwan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1217
ISBNs: 978-1-4704-4101-2 (print); 978-1-4704-4750-2 (online)
DOI: https://doi.org/10.1090/memo/1217
Published electronically: April 10, 2018
MSC: Primary 32V20, 32V30

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Table of Contents

Chapters

  • 1. Introduction and statement of the main results
  • 2. More properties of the phase $\varphi (x,y,s)$
  • 3. Preliminaries
  • 4. Semi-classical $\Box ^{(q)}_{b,k}$ and the characteristic manifold for $\Box ^{(q)}_{b,k}$
  • 5. The heat equation for the local operatot $\Box ^{(q)}_s$
  • 6. Semi-classical Hodge decomposition theorems for $\Box ^{(q)}_{s,k}$ in some non-degenerate part of $\Sigma$
  • 7. Szegö kernel asymptotics for lower energy forms
  • 8. Almost Kodaira embedding Theorems on CR manifolds
  • 9. Asymptotic expansion of the Szegö kernel
  • 10. Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR $S^1$ actions
  • 11. Szegő kernel asymptotics on some non-compact CR manifolds
  • 12. The proof of Theorem

Abstract

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 $q\in \left \{0,1,\ldots ,n-1\right \}$, 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, we prove that $\Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq 1$, 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, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ action.

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