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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| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, Rhode Island
American Mathematical Society
2018
|
| Vydanie: | 1 |
| Edícia: | Memoirs of the American Mathematical Society |
| Predmet: | |
| ISBN: | 9781470441012, 1470441012 |
| ISSN: | 0065-9266, 1947-6221 |
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| 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 \{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. |
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| AbstractList | 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. Let |
| Author | Hsiao, Chin-Yu |
| Author_xml | – sequence: 1 fullname: Hsiao, Chin-Yu |
| BackLink | https://cir.nii.ac.jp/crid/1130000796065752832$$DView record in CiNii |
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| Copyright | Copyright 2018 American Mathematical Society |
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| DEWEY | 516.36 |
| DOI | 10.1090/memo/1217 |
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| Discipline | Mathematics |
| EISBN | 1470447509 9781470447502 |
| EISSN | 1947-6221 |
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| Notes | Includes bibliographical references (p. 141-142) Volume 254, number 1217 (fifth of 5 numbers), July 2018 |
| OCLC | 1039082787 |
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| Snippet | Let 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... 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... |
| SourceID | askewsholts proquest nii ams |
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| SubjectTerms | CR submanifolds Functions of several complex variables Integral geometry |
| TableOfContents | Introduction and statement of the main results
--
More properties of the phase <inline-formula content-type="math/mathml"> φ ( x , y , s
) \varphi (x,y,s)
</inline-formula>
--
Preliminaries
--
Semi-classical <inline-formula content-type="math/mathml"> ◻ b ,
k ( q ) \Box
^{(q)}_{b,k} </inline-formula> and the characteristic manifold for <inline-formula
content-type="math/mathml"> ◻
b , k ( q )
\Box ^{(q)}_{b,k}
</inline-formula>
--
The heat equation for the local operatot <inline-formula content-type="math/mathml"> ◻ s ( q ) \Box ^{(q)}_s </inline-formula>
--
Semi-classical Hodge decomposition theorems for <inline-formula content-type="math/mathml">
◻
s , k ( q ) \Box ^{(q)}_{s,k} </inline-formula> in some
non-degenerate part of <inline-formula content-type="math/mathml">
Σ \Sigma
</inline-formula>
--
Szegö kernel asymptotics for lower energy forms
--
Almost Kodaira embedding Theorems on CR manifolds
--
Asymptotic expansion of the Szegö kernel
--
Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR <inline-formula
content-type="math/mathml"> S
1 S^1
</inline-formula> actions
--
Szegő kernel asymptotics on some non-compact CR manifolds
--
The proof of Theorem Cover -- Title page -- Chapter 1. Introduction and statement of the main results -- 1.1. Main results: Szegő kernel asymptotics for lower energy forms and almost Kodaira embedding Theorems on CR manifolds -- 1.2. Main results: Szegő kernel asymptotics -- 1.3. Main results: Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR ¹ actions -- Chapter 2. More properties of the phase ( , , ) -- Chapter 3. Preliminaries -- 3.1. Some standard notations -- 3.2. Set up and Terminology -- Chapter 4. Semi-classical \Box^{( )}_{ , } and the characteristic manifold for \Box^{( )}_{ , } -- Chapter 5. The heat equation for the local operatot \Box^{( )}_{ } -- 5.1. \Box^{( )}_{ } and the eikonal equation for \Box^{( )}_{ } -- 5.2. The transport equations for \Box^{( )}_{ } -- 5.3. Microlocal Hodge decomposition theorems for \Box^{( )}_{ } in -- 5.4. The tangential Hessian of ( , , ) -- Chapter 6. Semi-classical Hodge decomposition theorems for \Box^{( )}_{ , } in some non-degenerate part of Σ -- Chapter 7. Szegö kernel asymptotics for lower energy forms -- 7.1. Asymptotic upper bounds -- 7.2. Kernel of the spectral function -- 7.3. Szegö kernel asymptotics for lower energy forms -- Chapter 8. Almost Kodaira embedding Theorems on CR manifolds -- Chapter 9. Asymptotic expansion of the Szegö kernel -- Chapter 10. Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR ¹ actions -- 10.1. CR manifolds in projective spaces -- 10.2. Compact Heisenberg groups -- 10.3. Holomorphic line bundles over a complex torus -- Chapter 11. Szegő kernel asymptotics on some non-compact CR manifolds -- 11.1. The partial Fourier transform and the operator ^{( )}_{ , } -- 11.2. The small spectral gap property for \Box⁽⁰⁾_{ , } with respect to ⁽⁰⁾_{ , } 11.3. Szegő kernel asymptotics on Γ×\Real, where Γ=\Complexⁿ⁻¹ or Γ is a bounded strongly pseudoconvex domain in \Complexⁿ⁻¹ -- Chapter 12. The proof of Theorem 5.28 -- References -- Back Cover |
| Title | Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds |
| URI | https://www.ams.org/memo/1217/ https://cir.nii.ac.jp/crid/1130000796065752832 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=5501873 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781470447502 |
| Volume | 254 |
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