Singular Hilbert modules on Jordan-Kepler varieties

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.