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Resolution of Singularities for a Class of Hilbert Modules

Biswas, Shibananda and Misra, Gadadhar (2012) Resolution of Singularities for a Class of Hilbert Modules. In: INDIANA UNIVERSITY MATHEMATICS JOURNAL, 61 (3). pp. 1019-1050.

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Abstract

Let M be the completion of the polynomial ring C(z) under bar] with respect to some inner product, and for any ideal I subset of C (z) under bar], let I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule I] subset of M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors {p(i)(partial derivative/partial derivative(w) over bar (1),...,partial derivative/partial derivative(w) over bar (m)) K-I] (., w)vertical bar(w=0), 1 <= i <= t}, where K-I] is the reproducing kernel for the submodule 2] and p(1),..., p(t) is some minimal ``canonical set of generators'' for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the ``Rigidity Theorem'' using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of I]. Several examples are given to illustrate the explicit computation of these invariants.

Item Type: Journal Article
Additional Information: Copyright of this article is belongs to INDIANA UNIV MATH JOURNAL
Keywords: Hilbert module; reproducing kernel function; analytic Hilbert module; submodule; holomorphic Hermitian vector bundle; analytic sheaf
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 13 Sep 2013 09:11
Last Modified: 13 Sep 2013 09:11
URI: http://eprints.iisc.ac.in/id/eprint/47103

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