Asymptotics of torus equivariant Szegö kernel on a compact CR manifold

Abstract

For a compact CR manifold (X, T 1,0 X) of dimension 2n + 1, n ≥ 2, admitting a S 1 × T d action, if the lattice point (−p 1 , · · · , −p d ) ∈ Z d is a regular value of the associate CR moment map µ, then we establish the asymptotic expansion of the torus equivariant Szegő kernel Π (0) m,mp 1 ,··· ,mp d (x, y) as m → +∞ under certain assumptions of the positivity of Levi form and the torus action on Y := µ −1 (−p 1 , · · · , −p d ).

Asymptotics of G-equivariant Szegő kernels

“…Then 0 is the regular value of µ. By Theorem 1.10, we deduce the following which covers Shen's result [22] when X is strongly pseudoconvex.…”

The Bochner-Kodaira-Nakano formula for line bundles on CR manifolds with transversal CR R-action

“…The Bochner-Kodaira-Nakano formula [2] plays an important role in complex geometry, which is related to some classical results like Kodaira embedding theorem and Riemann-Roch-Hirzebruch formula. The counterpart of these results are well established for CR manifold with transversal CR S 1 -action [4,1] and related recent progress [11] [12] [14] [10] [6]. However, the Bochner-Kodaira-Nakano formula seems absent in this context.…”