Geometric Invariants for a Class of Submodules of Analytic Hilbert Modules Via the Sheaf Model

Abstract

Let $\Omega \subseteq {\mathbb {C}}^m$ be a bounded connected open set and ${\mathcal {H}} \subseteq {\mathcal {O}}(\Omega )$ be an analytic Hilbert module, i.e., the Hilbert space ${\mathcal {H}}$ possesses a reproducing kernel K, the polynomial ring $\mathbb C[{\varvec{z}}]\subseteq {\mathcal {H}}$ is dense and the point-wise multiplication induced by $p\in {\mathbb {C}}[{\varvec{z}}]$ is bounded on ${\mathcal {H}}$. We fix an ideal ${\mathcal {I}} \subseteq {\mathbb {C}}[{\varvec{z}}]$ generated by $p_1,\ldots ,p_t$ 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 ${\mathcal {O}}(\Omega )$ of holomorphic functions on $\Omega$ and hence is free. However, the subsheaf ${\mathcal {S}}^{\mathcal [{\mathcal {I}}]}$ associated to $[{\mathcal {I}}]$ is coherent and not necessarily locally free. Building on the earlier work of Biswas, Misra and Putinar (Journal fr die reine und angewandte Mathematik (Crelles Journal) 662:165–204, 2012), 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 $[{\mathcal {I}}] \subseteq {\mathcal {H}}$.