Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds

TL;DR

This paper studies the asymptotic behavior of the Szeg"o kernel on CR manifolds with line bundles, deriving Morse inequalities that relate geometric properties to spectral data, with applications to embedding problems.

Contribution

It provides new upper bounds for the Szeg"o kernel and establishes weak and strong Morse inequalities on CR manifolds, extending classical results to a broader setting.

Findings

Derived scaling upper bounds for Szeg"o kernel on (0, q)-forms.

Established weak Morse inequalities analogous to Demailly's.

Proved strong Morse inequalities with applications to manifold embeddings.

Abstract

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.