G-invariant Szegő kernel asymptotics and CR reduction

Chin Yu Hsiao, Rung Tzung Huang

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let (X, T1 , 0X) be a compact connected orientable CR manifold of dimension 2 n+ 1 with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group G action. Under certain natural assumptions about the group G action, we show that the G-invariant Szegő kernel for (0, q) forms is a complex Fourier integral operator, smoothing away μ- 1(0) and there is a precise description of the singularity near μ- 1(0) , where μ denotes the CR moment map. We apply our result to the case when X admits a transversal CR S1 action and deduce an asymptotic expansion for the mth Fourier component of the G-invariant Szegő kernel for (0, q) forms as m→ + ∞ and when q= 0 , we recover Xiaonan Ma and Weiping Zhang’s result about the existence of the G-invariant Bergman kernel for ample line bundles. As an application, we show that if m large enough, quantization commutes with reduction.

Original languageEnglish
Article number47
JournalCalculus of Variations and Partial Differential Equations
Volume60
Issue number1
DOIs
StatePublished - Feb 2021

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