article

Title: Resolution of singularities for a class of Hilbert modules

Authors: Shibananda Biswas and Gadadhar Misra

Issue: Volume 61 (2012), Issue 3, 1019-1050

Abstract:

$Let $\mathcal{M}$ be the completion of the polynomial ring $\mathbb{C}[\underline{z}]$ with respect to some inner product, and for any ideal $\mathcal{I}\subseteq\mathbb{C}[\underline{z}]$, let $[\mathcal{I}]$ be the closure of $\mathcal{I}$ in $\mathcal{M}$. For a homogeneous ideal $\mathcal{I}$, the joint kernel of the submodule $[\mathcal{I}]\subseteq\mathcal{M}$ is shown, after imposing some mild conditions on $\mathcal{M}$, to be the linear span of the set of vectors \[ \left\{p_i\left(\frac{\partial}{\partial\bar{w}_1},\dots,\frac{\partial}{\partial\bar{w}_m}\right)K_{[\mathcal{I}]}(\cdot,w)\Big|_{w=0}, 1\leq i\leq t\right\}, \] where $K_{[\mathcal{I}]}$ is the reproducing kernel for the submodule $[\mathcal{I}]$ and $p_1,\dots,p_t$ is some minimal "canonical set of generators" for the ideal $\mathcal{I}$. 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 $[\mathcal{I}]$. 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 $[\mathcal{I}]$. Several examples are given to illustrate the explicit computation of these invariants.$