AI 中文总结
研究主丛上殆复结构和凯勒度量的存在性与分类，用王的不变联络理论给出不变可积复结构分类的几何证明，用列维 - 奇维塔联络证明凯勒准则，应用结果表明上半平面和复射影空间中不变可积复结构唯一。
AI 中文摘要
本文研究主丛上殆复结构和凯勒度量的存在性与分类，尤其关注齐性情形。利用王的不变联络理论，直接给出了由比斯瓦斯和乌普迈尔建立的不变可积复结构分类的几何证明，避免了约旦三元系代数机制，从埃尔米特对称空间扩展到允许不变复结构的一般对称空间。还用列维 - 奇维塔联络给出了约翰逊凯勒准则的计算直接证明。最后将分类结果应用于上半平面和复射影空间的约化标架丛，表明这些情形下不变可积复结构是唯一的。
英文摘要
This work investigates the existence and classification of almost complex structures and Kähler metrics on principal bundles, with particular emphasis on the homogeneous setting. Using Wang's theory of invariant connections, we give a direct geometric proof of the classification of invariant integrable complex structures established by Biswas and Upmeier. This approach avoids the algebraic machinery of Jordan triple systems and extends the classification from Hermitian symmetric spaces to general symmetric spaces admitting invariant complex structures. We also present a computationally direct proof of Johnson's Kähler criterion using the Levi-Civita connection. Finally, we apply these classification results to the reduced frame bundles of the upper half-plane and complex projective spaces, showing that the invariant integrable complex structures in these cases are unique.
