A classical problem in Hermitian geometry, known as the equivalence problem, is to determine when two Hermitian holomorphic vector bundles are locally or globally equivalent. M. J. Cowen and R. G. Douglas related this problem to the issue of unitary equivalence of operators and introduced a broad and important class of operators denoted by $\mathbf{\mathcal{B}}_{n}^{m}(\Omega)$. They showed that the curvature and its covariant derivatives of the universal complex vector bundle associated with the operator in $\mathbf{\mathcal{B}}_{n}^{m}(\Omega)$  

serve as geometric invariants. Subsequently, C. Apostol, M. Martin, and others developed a more extensive approach to study this problem, providing a complete characterization of the equivalence of holomorphic mappings related to  Grassmann manifolds in the context of $C^{*}$-algebras. A natural question arises: in this broader non-commutative setting, what are the geometric quantities such as curvature and connection that correspond to these holomorphic mappings?We define the curvature and its covariant derivatives for extended holomorphic curves in the setting of $C^{*}$-algebras in the multivariable case, and establish connections between these geometric quantities and their counterparts in complex geometry. As applications, the obtained results are used to study the unitary classification and similarity classification of the Cowen-Douglas class $\mathbf{\mathcal{B}}_{n}^{m}(\Omega)$.

徐景，河北地质大学教师。主要研究复几何在线性算子理论中的应用，研究内容包括Cowen-Douglas算子与Hermitian全纯向量丛的结构与分类问题，包括利用几何不变量刻画算子的酉分类与相似分类；Hermitian全纯向量丛的旗结构、齐次性与弱齐次性；Cowen-Douglas 理论在C*代数中的拓展与应用；算子的相似分类与Corona问题等。主持国家自然科学基金青年项目，代表性科研论文发表于《Journal of Operator Theory》等学术期刊。