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

Abstract

Let $\Omega \subset \mathbb{C}^m$ be an open, connected and bounded set and $\mathcal{A}(\Omega)$ be a function algebra of holomorphic functions on $\Omega$. 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 $\mathcal{Z}\subset\Omega$ 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 $\mathbb{D}^m$.