G-invariant Szegö kernel asymptotics and CR reduction

2017/02/16 by Chin-Yu Hsiao, Hsiao, Chin-Yu, Rung-Tzung Huang +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1702.05012

openalex publication_date 2017/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X, T^1,0X) be a compact connected orientable CR manifold of dimension 2n+1 with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group action G. Under certain natural assumptions about the group action G, we show that the G-invariant Szegö 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 m-th Fourier component of the G-invariant Szegö kernel for (0,q) forms as m →+∞. As an application, we show that if m large enough, quantization commutes with reduction.

Citations

• Geometric quantization and multiplicities of group representations

Cited by

• Asymptotics of torus equivariant Szegő kernel on a compact CR manifold

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