 
Summary: THE CURVATURE INVARIANT OF A
HILBERT MODULE OVER C[z1, . . . , zd]
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
Abstract. A notion of curvature is introduced in multivariable operator theory,
that is, for commuting d tuples of operators acting on a common Hilbert space whose
"rank" is finite in an appropriate sense.
The curvature invariant is a real number in the interval [0, r] where r is the
rank, and for good reason it is desireable to know its value. For example, there
are significant and concrete consequences when it assumes either of the two extreme
values 0 or r. In the few simple cases where it can be calculated directly, it turns
out to be an integer. This paper addresses the general problem of computing this
invariant.
Our main result is an operatortheoretic version of the GaussBonnetChern for
mula of Riemannian geometry. The proof is based on an asymptotic formula which
expresses the curvature of a Hilbert module as the trace of a certain selfadjoint
operator. The Euler characteristic of a Hilbert module is defined in terms of the
algebraic structure of an associated finitely generated module over the algebra of
