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

Let Ω ⊆ C^m be a bounded connected open set and H⊆ O(Ω) 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 p<inf>1</inf>, … , p<inf>t</inf> and let [I] denote the completion of I in H. The sheaf S^H associated to analytic Hilbert module H is the sheaf O(Ω) of holomorphic functions on Ω 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<inf>[</inf><inf>I</inf><inf>]</inf> is a submanifold of codimension t, then there is a unique local decomposition for the kernel K<inf>[</inf><inf>I</inf><inf>]</inf> along the zero set that serves as a holomorphic frame for a vector bundle on V<inf>[</inf><inf>I</inf><inf>]</inf>. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule [I] ⊆ H.