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Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds


About this Title

Chin-Yu Hsiao

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 254, Number 1217
ISBNs: 978-1-4704-4101-2 (print); 978-1-4704-4750-2 (online)
DOI: https://doi.org/10.1090/memo/1217
Published electronically: April 10, 2018

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Table of Contents


Chapters

  • Chapter 1. Introduction and statement of the main results
  • Chapter 2. More properties of the phase $\varphi (x,y,s)$
  • Chapter 3. Preliminaries
  • Chapter 4. Semi-classical $\Box ^(q)_b,k$ and the characteristic manifold for $\Box ^(q)_b,k$
  • Chapter 5. The heat equation for the local operatot $\Box ^(q)_s$
  • Chapter 6. Semi-classical Hodge decomposition theorems for $\Box ^(q)_s,k$ in some non-degenerate part of $\Sigma $
  • Chapter 7. 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 $S^1$ actions
  • Chapter 11. Szegő kernel asymptotics on some non-compact CR manifolds
  • Chapter 12. The proof of Theorem 5.28

Abstract


Let be an abstract not necessarily compact orientable CR manifold of dimension , , and let be the -th tensor power of a CR complex line bundle over . Given , let be the Gaffney extension of Kohn Laplacian for forms with values in . For , let , where denotes the spectral measure of . In this work, we prove that , , , admit asymptotic expansions with respect to on the non-degenerate part of the characteristic manifold of , where is some kind of microlocal cut-off function. Moreover, we show that admits a full asymptotic expansion with respect to if has small spectral gap property with respect to and is -negligible away the diagonal with respect to . By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR action.

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