Szegő kernels and equivariant embedding theorems for CR manifolds

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb{R}$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier–Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish an $\mathbb{R}$-equivariant Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an $\mathbb{R}$-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasakian manifolds. As applications of our results, we obtain Torus equivariant Kodaira and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant Kodaira embedding theorems for complex manifolds.

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In memory of Professor Louis Boutet de Monvel

Hendrik Herrmann was partially supported by the CRC TRR 191: “Symplectic Structures in Geometry, Algebra and Dynamics”. He would like to thank the Mathematical Institute, Academia Sinica, and the School of Mathematics and Statistics, Wuhan University, for hospitality, a comfortable accommodation and financial support during his visits in January and March-April, respectively. Chin-Yu Hsiao was partially supported by Taiwan Ministry of Science and Technology project 106-2115-M-001-012 and Academia Sinica Career Development Award and he would like to thank Professor Homare Tadano for useful discussion in this work. Xiaoshan Li was supported by National Natural Science Foundation of China (Grant No. 11871380, 11501422).

Received 25 November 2019

Accepted 12 July 2020

Published 6 September 2022