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Szegő Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds
 
Chin-Yu Hsiao Academia Sinica, Taipei, Taiwan
Szego Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds
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
Szego Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds
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Szegő Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds
Chin-Yu Hsiao Academia Sinica, Taipei, Taiwan
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
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2542018; 140 pp
    MSC: 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.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2542018; 140 pp
MSC: 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.

  • 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
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