Similarity invariants of essentially normal Cowen-Douglas operators and Chern polynomials

In this paper, we systematically study a class of essentially normal operators by using the geometry method from the Cowen–Douglas theory and prove a Brown–Douglas–Fillmore theorem in the Cowen–Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.