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

Let $$\Omega \subseteq {\mathbb {C}}^m$$ Ω ⊆ C m be a bounded connected open set and $${\mathcal {H}} \subseteq {\mathcal {O}}(\Omega )$$ H ⊆ O ( Ω ) be an analytic Hilbert module, i.e., the Hilbert space $${\mathcal {H}}$$ H possesses a reproducing kernel K, the polynomial ring $$\mathbb C[{\varvec{z}}]\subseteq {\mathcal {H}}$$ C [ z ] ⊆ H is dense and the point-wise multiplication induced by $$p\in {\mathbb {C}}[{\varvec{z}}]$$ p ∈ C [ z ] is bounded on $${\mathcal {H}}$$ H . We fix an ideal $${\mathcal {I}} \subseteq {\mathbb {C}}[{\varvec{z}}]$$ I ⊆ C [ z ] generated by $$p_1,\ldots ,p_t$$ p 1 , … , p t and let $$[{\mathcal {I}}]$$ [ I ] denote the completion of $${\mathcal {I}}$$ I in $$\mathcal H$$ H . The sheaf $${\mathcal {S}}^{\mathcal {H}}$$ S H associated to analytic Hilbert module $${\mathcal {H}}$$ H is the sheaf $${\mathcal {O}}(\Omega )$$ O ( Ω ) of holomorphic functions on $$\Omega $$ Ω and hence is free. However, the subsheaf $${\mathcal {S}}^{\mathcal [{\mathcal {I}}]}$$ S [ I ] associated to $$[{\mathcal {I}}]$$ [ 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 $$V_{[{\mathcal {I}}]}$$ V [ I ] is a submanifold of codimension t, then there is a unique local decomposition for the kernel $$K_{[{\mathcal {I}}]}$$ K [ I ] along the zero set that serves as a holomorphic frame for a vector bundle on $$V_{[{\mathcal {I}}]}$$ V [ I ] . The complex geometric invariants of this vector bundle are also unitary invariants for the submodule $$[{\mathcal {I}}] \subseteq {\mathcal {H}}$$ [ I ] ⊆ H .