eBook ISBN:  9781470447502 
Product Code:  MEMO/254/1217.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 
eBook ISBN:  9781470447502 
Product Code:  MEMO/254/1217.E 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $46.80 

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 \(2n1\), \(n\geqslant 2\), and let \(L^k\) be the \(k\)th tensor power of a CR complex line bundle \(L\) over \(X\). Given \(q\in \{0,1,\ldots ,n1\}\), 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 nondegenerate part of the characteristic manifold of \(\Box ^{(q)}_{b,k}\), where \(F_k\) is some kind of microlocal cutoff 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.

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. Semiclassical $\Box ^{(q)}_{b,k}$ and the characteristic manifold for $\Box ^{(q)}_{b,k}$

5. The heat equation for the local operatot $\Box ^{(q)}_s$

6. Semiclassical Hodge decomposition theorems for $\Box ^{(q)}_{s,k}$ in some nondegenerate 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 noncompact CR manifolds

12. The proof of Theorem


Additional Material

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Let \(X\) be an abstract not necessarily compact orientable CR manifold of dimension \(2n1\), \(n\geqslant 2\), and let \(L^k\) be the \(k\)th tensor power of a CR complex line bundle \(L\) over \(X\). Given \(q\in \{0,1,\ldots ,n1\}\), 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 nondegenerate part of the characteristic manifold of \(\Box ^{(q)}_{b,k}\), where \(F_k\) is some kind of microlocal cutoff 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. Semiclassical $\Box ^{(q)}_{b,k}$ and the characteristic manifold for $\Box ^{(q)}_{b,k}$

5. The heat equation for the local operatot $\Box ^{(q)}_s$

6. Semiclassical Hodge decomposition theorems for $\Box ^{(q)}_{s,k}$ in some nondegenerate 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 noncompact CR manifolds

12. The proof of Theorem