# Singular Hilbert modules on Jordan-Kepler varieties

## TL;DR

This paper investigates submodules of analytic Hilbert modules over Jordan-Kepler varieties, introducing a novel higher rank blow-up process to resolve singularities and establish submodule rigidity.

## Contribution

It develops a new Jordan algebraic determinant-based blow-up method for resolving singularities in higher rank varieties, advancing the understanding of submodule structure.

## Key findings

- Resolved singularities using a new higher rank blow-up process.
- Proved rigidity of submodules vanishing on the singular set.
- Extended analysis to varieties of arbitrary rank.

## Abstract

We study submodules of analytic Hilbert modules defined over certain algebraic varieties in bounded symmetric domains, the so-called Jordan-Kepler varieties $V_\ell$ of arbitrary rank $\ell.$ For $\ell>1$ the singular set of $V_\ell$ is not a complete intersection. Hence the usual monoidal transformations do not suffice for the resolution of the singularities. Instead, we describe a new higher rank version of the blow-up process, defined in terms of Jordan algebraic determinants, and apply this resolution to obtain the rigidity of the submodules vanishing on the singular set.