Hermitian geometry and holomorphic curves on $C^{*}$-algebras
Streszczenie
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 $\mathcal {B}_{n}^{m}(\varOmega )$. They showed that the curvature and its covariant derivatives of the universal complex vector bundle associated with an operator in $\mathcal {B}_{n}^{m}(\varOmega )$ 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?
This paper aims to define the curvature and its covariant derivatives for extended holomorphic curves in the setting of $C^{*}$-algebras in the multivariable case, and to establish connections between these geometric quantities and their counterparts in complex geometry. As applications, the results obtained are used to study the unitary classification and similarity classification of the Cowen–Douglas class $\mathcal {B}_{n}^{m}(\varOmega )$.
