On the Szeg\"o kernel of Cartan-Hartogs domains

TL;DR

This paper computes the Szeg"o kernel for disk bundles over Cartan-Hartogs domains, extending methods from compact symmetric spaces to noncompact settings, revealing new insights into their complex geometric structure.

Contribution

It provides the first explicit computation of the Szeg"o kernel for disk bundles over Cartan-Hartogs domains, adapting techniques from compact symmetric spaces.

Findings

Explicit Szeg"o kernel formula for Cartan-Hartogs domains

Demonstrates the disk bundle structure as iterated bundles

Extends compact symmetric space methods to noncompact cases

Abstract

Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szeg\"o kernel of the disk bundle over a noncompact K\"ahler manifold. In particular we compute the Szeg\"o kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szeg\"o kernel of the disk bundle over an Herimitian symmetric space of compact type.