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

Let (formula presented) ⊆ Cm be a bounded connected open set and H ⊆O(formula presented) be an analytic Hilbert module, i.e., the Hilbert space H possesses a reproducing kernel K, the polynomial ring C[z] ⊆ H is dense and the point-wise multiplication induced by p ∈ C[z] is bounded on H. We fix an ideal I ⊆ C[z] generated by p1, pt and let [I] denote the completion of I in H. The sheaf SH associated to analytic Hilbert module H is the sheaf O(formula presented) of holomorphic functions on (formula presented) and hence is free. However, the subsheaf S[I] associated to [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[I] is a submanifold of codimension t, then there is a unique local decomposition for the kernel K[I] along the zero set that serves as a holomorphic frame for a vector bundle on V[I]. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule [I] ⊆ H.