2022/10/30 by Shibananda Biswas, Biswas, Shibananda, Gadadhar Misra +3
Computer Science · Mathematics · #32A10 #32A36 #32A38 #47B13 #47B32 #47B35 #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Functional Analysis (math.FA) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2210.16912
openalex publication_date 2022/10/30 · openalex created_date 2022/11/06 · openalex updated_date 2026/07/28
Let Ω⊆ \mathbb Cm be a bounded connected open set and \mathcal H ⊆ \mathcal O(Ω) be an analytic Hilbert module, i.e., the Hilbert space \mathcal H possesses a reproducing kernel K, the polynomial ring \mathbb C[\boldsymbolz]⊆ \mathcal H is dense and the point-wise multiplication induced by p∈ \mathbb C[\boldsymbolz] is bounded on \mathcal H. We fix an ideal \mathcal I ⊆ \mathbb C[\boldsymbolz] generated by p1,…,pt 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(Ω) of holomorphic functions on Ω 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 \citeBMP, 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] ⊆ \mathcal H.