Szegő Kernel Asymptotics for High Power of CR Line Bundles and Kodaira Embedding Theorems on CR Manifolds.
- 1st ed.
- 1 online resource (154 pages)
- Memoirs of the American Mathematical Society Series ; v.254 .
- Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 1. Introduction and statement of the main results -- 1.1. Main results: Szegő kernel asymptotics for lower energy forms and almost Kodaira embedding Theorems on CR manifolds -- 1.2. Main results: Szegő kernel asymptotics -- 1.3. Main results: Szegő kernel asymptotics and Kodairan embedding theorems on CR manifolds with transversal CR ¹ actions -- Chapter 2. More properties of the phase ( , , ) -- Chapter 3. Preliminaries -- 3.1. Some standard notations -- 3.2. Set up and Terminology -- Chapter 4. Semi-classical \Box^_ and the characteristic manifold for \Box^_ -- Chapter 5. The heat equation for the local operatot \Box^_ -- 5.1. \Box^_ and the eikonal equation for \Box^_ -- 5.2. The transport equations for \Box^_ -- 5.3. Microlocal Hodge decomposition theorems for \Box^_ in -- 5.4. The tangential Hessian of ( , , ) -- Chapter 6. Semi-classical Hodge decomposition theorems for \Box^_ in some non-degenerate part of Σ -- Chapter 7. Szegö kernel asymptotics for lower energy forms -- 7.1. Asymptotic upper bounds -- 7.2. Kernel of the spectral function -- 7.3. 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 ¹ actions -- 10.1. CR manifolds in projective spaces -- 10.2. Compact Heisenberg groups -- 10.3. Holomorphic line bundles over a complex torus -- Chapter 11. Szegő kernel asymptotics on some non-compact CR manifolds -- 11.1. The partial Fourier transform and the operator ^_ -- 11.2. The small spectral gap property for \Box⁽⁰⁾_ with respect to ⁽⁰⁾_. 11.3. Szegő kernel asymptotics on Γ×\Real, where Γ=\Complexⁿ⁻¹ or Γ is a bounded strongly pseudoconvex domain in \Complexⁿ⁻¹ -- Chapter 12. The proof of Theorem 5.28 -- References -- Back Cover.
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 \, let \Box ^_ be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For \lambda \geq 0, let \Pi ^_ :=E((-\infty ,\lambda ]), where E denotes the spectral measure of \Box ^_. In this work, the author proves that \Pi ^_}F^*_k, F_k\Pi ^_}F^*_k, N_0\geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of \Box ^_, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_k\Pi ^_F^*_k admits a full asymptotic expansion with respect to k if \Box ^_ has small spectral gap property with respect to F_k and \Pi^_ 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.