Szego Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds
Seiten
2018
American Mathematical Society (Verlag)
978-1-4704-4101-2 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-4101-2 (ISBN)
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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 $L^k$.
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 $X$ 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.
Introduction and statement of the main results;
More properties of the phase $/varphi (x,y,s)$; Preliminaries;
Semi-classical $/Box ^{(q)}_{b,k}$ and the characteristic manifold for $/Box ^{(q)}_{b,k}$;
The heat equation for the local operatot $/Box ^{(q)}_s$;
Semi-classical Hodge decomposition theorems for $/Box ^{(q)}_{s,k}$ in some non-degenerate part of $/Sigma $; Szego kernel asymptotics for lower energy forms;
Almost Kodaira embedding Theorems on CR manifolds;
Asymptotic expansion of the Szego kernel;
Szego kernel asymptotics and Kodairan embedding Theorems on CR manifolds with transversal CR $S^1$ actions;
Szego kernel asymptotics on some non-compact CR manifolds;
The proof of Theorem 5.28;
References.
| Erscheinungsdatum | 07.10.2018 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 242 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-4101-2 / 1470441012 |
| ISBN-13 | 978-1-4704-4101-2 / 9781470441012 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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