Remarks on asymptotic isometric embeddings of conic transforms for torus actions

TL;DR

This paper investigates asymptotic isometric embeddings of conic transforms associated with torus actions on Hodge manifolds, utilizing equivariant Szegő projectors to understand their geometric and representation-theoretic properties.

Contribution

It introduces a new approach to asymptotic embeddings of conic transforms using equivariant Szegő projectors in the context of Hamiltonian torus actions on Hodge manifolds.

Findings

Established asymptotic embeddings for conic transforms.

Connected geometric loci to representation theory via Szegő projectors.

Provided conditions under which embeddings are asymptotically isometric.

Abstract

Consider a Hodge manifold and assume that a torus acts on it in a Hamiltonian and holomorphic manner and that this action linearizes on a given quantizing line bundle. Inside the dual of the line bundle one can define the circle bundle, which is a strictly pseudoconvex CR manifold. Then, there is an associated unitary representation on the Hardy space of the circle bundle. Under suitable assumptions on the moment map, we consider certain loci in unit circle bundle, naturally associated to a ray through an irreducible weight. Their quotients are called conic transforms. We introduce maps which are asymptotic embeddings of conic transforms making use of the corresponding equivariant Szeg\H{o} projector.