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

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...

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Bibliographic Details
Main Author: Hsiao, Chin-Yu
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2018
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
CR submanifolds
Functions of several complex variables
Integral geometry
ISBN:9781470441012, 1470441012
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Summary: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.
Bibliography:Includes bibliographical references (p. 141-142)
Volume 254, number 1217 (fifth of 5 numbers), July 2018
ISBN:9781470441012
1470441012
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1217