Szegő kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds

Abstract

Let $X$ be an orientable compact Levi-flat CR manifold and let $L$ be a positive CR complex line bundle over $X$. We prove that certain microlocal conjugations of the associated Szegő kernel admits an asymptotic expansion with respect to high powers of $L$. As an application, we give a Szegő kernel proof of the Kodaira type embedding theorem on Levi-flat CR manifolds due to Ohsawa and Sibony.

45 pages; expanded version including background on Levi-flat manifolds and several comments on the nature of the projector Π_k; v.2 is a final update to agree with the published paper