Szegő kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds - Hsiao, Chin-Yu (Memoirs of the American Mathematical Society, 2018)
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,… ,n-1\}, let \Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For λ ≥ 0, let Π ^{(q)}_{k,≤ λ} :=E((-∞ ,λ ]), where E denotes the spectral measure of \Box ^{(q)}_{b,k}. In this work, the author proves that Π ^{(q)}_{k,≤ k^{-N_0}}F^*_k, F_kΠ ^{(q)}_{k,≤ k^{-N_0}}F^*_k, N_0≥ 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_kΠ ^{(q)}_{k,≤ 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 Π^{(q)}_{k,≤ 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.