A local theory for operator tuples in the Cowen-Douglas class

We present a local theory for a commuting m-tuple S=(S1,S2,⋯,Sm) of Hilbert space operators lying in the Cowen-Douglas class. By representing S on a Hilbert module M consisting of vector-valued holomorphic functions over Cm, we identify and study the localization of S on an analytic hyper-surface in Cm. We completely determine unitary equivalence of the localization and relate it to geometric invariants of the Hermitian holomorphic vector bundle associated to S. It turns out that the localization coincides with an important class of quotient Hilbert modules, and our result concludes its classification problem in full generality.