The curvature invariant in multivariable operator theory I, II
by
Bill Arveson
University of California at Berkeley

The Gauss Bonnet formula asserts that the integral of the Gaussian curvature over a compact oriented Riemann surface is an integer, namely the alternating sum of the three Betti numbers of the surface...the Euler characteristic of the surface. This remarkable theorem was generalized in the late forties to higher dimensional manifolds, most notably by S.-S. Chern.

After reviewing these geometric issues for analysts, we discuss some basic constructions with commuting sets of operators acting on a common Hilbert space H. Such an H is a module over the algebra of complex polynomials in the corresponding number of variables, and is called a Hilbert module. When it is of "finite rank" in an appropriate sense it is possible to define a numerical invariant K(H) which is analogous to the average curvature of a manifold. This invariant is a nonnegative real number, it has some interesting properties, it is not easy to compute, and it has a tendency to be an integer.

We also introduce an operator-theoretic analogue of the Euler characteristic. This invariant depends only on the linear algebra of a certain complex vector space acted upon by a commuting set of linear transformations...and which is finitely generated in the usual algebraic sense. The Euler characteristic is a nonnegative integer and there is well-established technology (going back to Hilbert) for computing it. The central result of these lectures is that for "graded" Hilbert modules, K(H) agrees with the Euler characteristic of H. Time permitting, we discuss some natural examples and some applications.

Date received: May 4, 1999