Asymptotics of torus equivariant Szeg\H{o} kernel on a compact CR manifold

TL;DR

This paper derives the asymptotic expansion of the torus equivariant Szeg ext{"o} kernel on a compact CR manifold with a torus action, under regularity and positivity conditions, extending understanding of kernel behavior in geometric analysis.

Contribution

It establishes the asymptotic expansion of the torus equivariant Szeg ext{"o} kernel on CR manifolds with torus symmetry, under regularity and positivity assumptions, advancing the analysis of geometric kernels.

Findings

Asymptotic expansion of Szeg ext{"o} kernel derived

Conditions for regularity and positivity established

Results applicable to CR manifolds with torus actions

Abstract

For a compact CR manifold $(X,T^{1,0}X)$ of dimension $2n+1$, $n\geq 2$, admitting a $S^1\times T^d$ action, if the lattice point $(-p_1,\cdots,-p_d)\in\mathbb{Z}^{d}$ is a regular value of the associate CR moment map $\mu$, then we establish the asymptotic expansion of the torus equivariant Szeg\H{o} kernel $\Pi^{(0)}_{m,mp_1,\cdots,mp_d}(x,y)$ as $m\to +\infty$ under certain assumptions of the positivity of Levi form and the torus action on $Y:=\mu^{-1}(-p_1,\cdots,-p_d)$.