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 \([{\mathcal {I}}] \subseteq {\mathcal {H}}\).