The regular quantizations of certain holomorphic bundles

TL;DR

This paper investigates conditions under which certain holomorphic bundles over Kähler manifolds admit regular quantizations, revealing that some projective bundles over Fano manifolds do so only if the base is a complex projective space, and constructs balanced metrics on bundles over the Riemann sphere.

Contribution

It provides necessary and sufficient conditions for regular quantizations of holomorphic bundles and characterizes when projective bundles over Fano manifolds admit such quantizations.

Findings

Regular quantizations characterized by Bergman function coefficients.

Projective bundles over Fano manifolds admit regular quantizations only over complex projective spaces.

Balanced metrics constructed on bundles over the Riemann sphere.

Abstract

In this paper, we study the regular quantizations of K\"{a}hler manifolds by using the first two coefficients of Bergman function expansions. Firstly, we obtain sufficient and necessary conditions for certain Hermitian holomorphic vector bundles and their ball subbundles to be regular quantizations. Secondly, we obtain that some projective bundles over the Fano manifolds $M$ admit regular quantizations if and only if $M$ are biholomorphically isomorphism to the complex projective spaces. Finally, we obtain the balanced metrics on certain Hermitian holomorphic vector bundles and their ball subbundles over the Riemann sphere.