The curvature invariant of a Hilbert module over C[z_1,...,z_d]

TL;DR

This paper introduces a curvature invariant in multivariable operator theory, establishes a Gauss-Bonnet-Chern type theorem for graded Hilbert modules, and computes invariants for examples related to complex projective varieties.

Contribution

It develops a new notion of curvature for multivariable Hilbert modules and proves an analogue of the Gauss-Bonnet-Chern theorem in this setting.

Findings

Curvature invariant defined for graded Hilbert modules.

Gauss-Bonnet-Chern type theorem established.

Explicit computations for examples in complex projective space.

Abstract

A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established for graded (contractive) Hilbert modules over the complex polynomial algebra in d variables, d=1,2,3,.... The curvature invariant, Euler characteristic, and degree are computed for some explicit examples based on varieties in (multidimensional) complex projective space, and applications are given to the structure of graded ideals in C[z_1,...,z_d] and to the existence of "inner sequences" for closed submodules of the free Hilbert module H^2(C^d).