Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension

Abstract

In this paper, we consider a CR manifold X of high codimension and a CR line bundle L over X. For each k ∈ ℕ, let L^k denote the k-th power of L. We establish that the spectral projection Π_{k,≤ kμ_k} := 1_{(−∞, kμ_k]}(Box_b,k) converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions, where Box_b,k denotes the Kohn Laplacian with values in L^k. Moreover, when Box_b,k possesses a local spectral gap, we show that the Szegő projection Π_k converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions. We also provide an explicit formula for the Szegő projection on the high codimension Heisenberg group. Furthermore, we consider a high codimension CR manifold X equipped with a transversal and CR ℝ^m-action. In this setting, we prove that the partial Szegő projection Π_{k,(ℝ^mcurvearrowright,≤ kδ)} converges to the weighted partial Szegő projection on the high codimension Heisenberg group in the 𝒞^∞ topology. As an application of our results, we establish a Kodaira embedding theorem for high codimension CR manifolds with a transversal and CR ℝ^m-action.

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BibTeX RIS

BibTeX

@misc{hsiao2026semiclassical,
  title = {{Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension}},
  author = {Chin-Yu Hsiao and Chin Jui Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25255},
  eprint = {2610.25255},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25255},
}

RIS

TY  - UNPB
TI  - Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension
AU  - Chin-Yu Hsiao
AU  - Chin Jui Yang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25255
UR  - https://arxiv.org/abs/2610.25255
AB  - In this paper, we consider a CR manifold X of high codimension and a CR line bundle L over X. For each k ∈ ℕ, let L^k denote the k-th power of L. We establish that the spectral projection Π_k,≤ kμ_k := 1_(−∞, kμ_k](Box_b,k) converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions, where Box_b,k denotes the Kohn Laplacian with values in L^k. Moreover, when Box_b,k possesses a local spectral gap, we show that the Szegő projection Π_k converges to the weighted Szegő projection on the high codimension Heisenberg group in the sense of distributions. We also provide an explicit formula for the Szegő projection on the high codimension Heisenberg group. Furthermore, we consider a high codimension CR manifold X equipped with a transversal and CR ℝ^m-action. In this setting, we prove that the partial Szegő projection Π_k,(ℝ^mcurvearrowright,≤ kδ) converges to the weighted partial Szegő projection on the high codimension Heisenberg group in the 𝒞^∞ topology. As an application of our results, we establish a Kodaira embedding theorem for high codimension CR manifolds with a transversal and CR ℝ^m-action.
ER  -

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