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

TL;DR

This paper investigates the unitary invariants of quotient Hilbert modules derived from submodules vanishing along complex analytic sets, linking algebraic properties to geometric invariants of vector bundles, with applications to weighted Bergman modules.

Contribution

It provides a complete characterization of unitary equivalence of quotient modules along smooth complex analytic sets and relates these to geometric invariants of Hermitian holomorphic vector bundles.

Findings

Complete determination of unitary equivalence for quotient modules

Relation of invariants to Hermitian holomorphic vector bundles

Characterization of weighted Bergman modules via quotient modules

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 $\mathbb{D}^m$.