Equivalence of quotient Hilbert modules -- II

For any open, connected and bounded set $Ω\subseteq \mathbb C^m$, let $\mathcal A$ be a natural function algebra consisting of functions holomorphic on $Ω$. Let $\mathcal M$ be a Hilbert module over the algebra $\mathcal A$ and $\mathcal M_0\subseteq \mathcal M$ be the submodule of functions vanishing to order $k$ on a hypersurface $\mathcal Z \subseteq Ω$. Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module $\mathcal Q = \mathcal M \ominus \mathcal M_0$ 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 certain homogeneous Hilbert modules over the bi-disc algebra.