Resolution of Singularities for a Class of Hilbert Modules

• Published in: Indiana University mathematics journal 2012-01, Vol.61 (3), p.1019-1050
• Main Authors: Biswas, Shibananda, Misra, Gadadhar
• Format: Article
• Language: English
• Subjects: Algebra; Analytic functions; Curvature; Inner products; Mathematical functions; Mathematical rings; Mathematical theorems; Mathematical vectors; Polynomials
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2012.61.4633

Let M be the completion of the polynomial ring C[z] with respect to some inner product, and for any ideal I ⊆ C[z], let [I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule [I] ⊆ M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors $\left\{{\mathrm{p}}_{\mathrm{i}}(\frac{\partial }{\partial {\overline{\mathrm{w}}}_{1}},\ldots,\frac{\partial }{\partial {\overline{\mathrm{w}}}_{\mathrm{m}}}){\mathrm{K}}_{\left[\mathcal{I}\right]}(\cdot,\mathrm{w}){|}_{\mathrm{w}=0},1\le \mathrm{i}\le \mathrm{t}\right\}$, where K[I] is the reproducing kernel for the submodule [I] and p1,...,pt is some minimal "canonical set of generators" for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the "Rigidity Theorem" using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form [I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of [I]. Several examples are given to illustrate the explicit computation of these invariants.