Geometric invariants for a class of submodules of analytic Hilbert modules via the sheaf model

Abstract

Let $\hat{I}\text{c◯}\subseteq {\mathbb{C}}^m$ be a bounded connected open set and $\mathcal{H}\subseteq \mathcal{O}(\hat{I}\text{c◯})$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal{H}$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb{C}[\mathbf{z}]\subseteq \mathcal{H}$ is dense and the point-wise multiplication induced by $p\in \mathbb{C}[\mathbf{z}]$ is bounded on $\mathcal{H}$. We fix an ideal $\mathcal{I}\subseteq \mathbb{C}[\mathbf{z}]$ generated by $p_1,\dots ,p_t$ and let $[\mathcal{I}]$ denote the completion of $\mathcal{I}$ in $\mathcal{H}$. The sheaf ${\mathcal{S}}^{\mathcal{H}}$ associated to analytic Hilbert module $\mathcal{H}$ is the sheaf $\mathcal{O}(\hat{I}\text{c◯})$ of holomorphic functions on $\hat{I}\text{c◯}$ and hence is free. However, the subsheaf ${\mathcal{S}}^{[\mathcal{I}]}$ associated to $[\mathcal{I}]$ is coherent and not necessarily locally free. Building on the earlier work of \cite{BMP}, we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set $V_{[\mathcal{I}]}$ is a submanifold of codimension $t$, then there is a unique local decomposition for the kernel $K_{[\mathcal{I}]}$ along the zero set that serves as a holomorphic frame for a vector bundle on $V_{[\mathcal{I}]}$. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule $[\mathcal{I}]\subseteq \mathcal{H}$.

To appear in Complex Analysis and Operator Theory