  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.

Complex Variables