On unitary invariants of quotient Hilbert modules along smooth complex analytic sets

Let $\mathrm{\Omega }\subset {\mathbb{C}}^m$ be an open, connected and bounded set and $\mathcal{A}(\mathrm{\Omega })$ be a function algebra of holomorphic functions on $\mathrm{\Omega }$. In this article we study quotient Hilbert modules obtained from submodules, consisting of functions in $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subset \mathrm{\Omega }$ of codimension at least $2$, of a quasi-free Hilbert module, $\mathcal{M}$. Our motive is to investigate unitary invariants of such quotient modules. We completely determine unitary equivalence of aforementioned quotient modules and relate it to geometric invariants of a Hermitian holomorphic vector bundles. Then, as an application, we characterize unitary equivalence classes of weighted Bergman modules over $\mathcal{A}({\mathbb{D}}^m)$ in terms of those of quotient modules arising from the submodules of functions vanishing to order $2$ along the diagonal in ${\mathbb{D}}^m$.