In this paper, we study the regular quantizations of Kähler 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.

中文翻译：

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某些全纯丛的正则量化

在本文中，我们通过使用 Bergman 函数展开的前两个系数来研究 Kähler 流形的正则量化。首先，我们获得了某些 Hermitian 全纯向量丛及其球子丛是正则量化的充分必要条件。其次，我们得到 Fano 流形 M 上的一些射影丛允许正则量化当且仅当 M 是复射影空间的双全同同构。最后，我们在黎曼球面上获得了某些 Hermitian 全纯向量丛及其球子丛的平衡度量。