Szego kernel, regular quantizations and spherical CR-structures

TL;DR

This paper computes the Szego kernel for certain complex manifolds, constructs examples of pseudoconvex domains with special properties, and explores the implications of spherical CR-structures on homogeneous Hodge manifolds.

Contribution

It provides explicit Szego kernel formulas, constructs new pseudoconvex domains with vanishing log-terms, and proves a characterization of projective spaces via CR-structures.

Findings

Szego kernel computed for unit circle bundles over compact Kaehler manifolds.

Constructed infinite families of pseudoconvex domains with vanishing log-terms.

Established that spherical CR-structures imply biholomorphism to projective space for homogeneous Hodge manifolds.

Abstract

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by M. Englis and G. Zhang for Hermitian symmetric spaces of compact type.