The curvature of a Hilbert module over ℂ[z₁, … , z_d]

Abstract

A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over ℂ[z₁, … , z_d] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring. This result is a higher dimensional operator-theoretic counterpart of the Gauss–Bonnet formula which expresses the average Gaussian curvature of a compact oriented Riemann surface as the alternating sum of the Betti numbers of the surface, and it solves the problem of calculating the curvature of graded Hilbert modules. The proof of that result is based on an asymptotic formula which expresses the curvature of a Hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the Euler characteristic.