Curvature and the Second fundamental form in classifying quasi-homogeneous holomorphic curves and operators in the Cowen-Douglas class

TL;DR

This paper classifies quasi-homogeneous operators in the Cowen-Douglas class using complex geometric invariants like curvature and second fundamental form, extending previous classifications and addressing similarity questions.

Contribution

It provides new canonical models for quasi-homogeneous operators based on geometric invariants, extending prior work on homogeneous operators.

Findings

Operators are irreducible and often strongly irreducible.

Classification theorems for unitary and invertible equivalence.

Confirmed the equality of topological and algebraic K-groups for these operators.

Abstract

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen-Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen-Douglas class obtained recently by the last author and A. Kor\'{a}nyi. Specifically, the complex geometric invariants used for our classification are the curvature and the second fundamental forms inherent in the definition of a quasi-homogeneous operator. We show that these operators are irreducible and determine when they are strongly irreducible. Applications include the equality of the topological and algebraic K-group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos on similarity for these operators.