Asymptotics of torus equivariant Szegő kernel on a compact CR manifold

Abstract

For a compact CR manifold $(X,T^{1,0}X)$ of dimension $2n+1$, $n\ge 2$, 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 Szegő kernel ${\Pi }_{m,m{p}_1,\cdots ,m{p}_d}^{(0)}(x,y)$ as $m\to +\infty$ under certain assumptions of the positivity of Levi form and the torus action on $Y:={\mu }^{-1}(-p_1,\cdots ,-p_d)$.

37 pages