Eqivalence of quotient Hilbert modules - II

Abstract

For any open, connected and bounded set Ω ⊆ C^m, let A be a natural function algebra consisting of functions holomorphic on Ω. Let M be a Hilbert module over the algebra A and M₀ ⊆ M be the submodule of functions vanishing to order k on a hypersurface Z ⊆ Ω. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module Q = M⊝ M₀ in the case of k = 2. In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise homogeneous modules over the bi-disc algebra.