
eBook ISBN: | 978-1-4704-4750-2 |
Product Code: | MEMO/254/1217.E |
List Price: | $78.00 |
MAA Member Price: | $70.20 |
AMS Member Price: | $46.80 |

eBook ISBN: | 978-1-4704-4750-2 |
Product Code: | MEMO/254/1217.E |
List Price: | $78.00 |
MAA Member Price: | $70.20 |
AMS Member Price: | $46.80 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 254; 2018; 140 ppMSC: Primary 32
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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Table of Contents
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Chapters
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1. Introduction and statement of the main results
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2. More properties of the phase $\varphi (x,y,s)$
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3. Preliminaries
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4. Semi-classical $\Box ^{(q)}_{b,k}$ and the characteristic manifold for $\Box ^{(q)}_{b,k}$
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5. The heat equation for the local operatot $\Box ^{(q)}_s$
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6. Semi-classical Hodge decomposition theorems for $\Box ^{(q)}_{s,k}$ in some non-degenerate part of $\Sigma $
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7. Szegö kernel asymptotics for lower energy forms
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8. Almost Kodaira embedding Theorems on CR manifolds
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9. Asymptotic expansion of the Szegö kernel
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10. Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR $S^1$ actions
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11. Szegő kernel asymptotics on some non-compact CR manifolds
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12. The proof of Theorem
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Additional Material
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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.
-
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