A refined notion of curvature for a linear system of Hermitian vector spaces,
in the sense of Grothendieck, leads to the unitary classification of a large
class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules,
for which the localizations are of finite (but not constant) dimension, of an
analytic function space with a reproducing kernel. The correspondence between
analytic Hilbert modules of constant rank and holomorphic Hermitian bundles on
domains of $\mathbb C^n$ due to Cowen and Douglas, as well as a natural
analytic localization technique derived from the Hochschild cohomology of
topological algebras play a major role in the proofs. A series of concrete
computations, inspired by representation theory of linear groups, illustrate
the abstract concepts of the paper.