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 operator-theoretic version
of the Gauss-Bonnet-Chern formula 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 self-adjoint 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 complex polynomials
$\Bbb C[z_1,\dots,z_d]$, and
the result is
that these two numbers are the same for
graded Hilbert modules. Thus the
curvature of such a Hilbert module is
an integer; and since there are
standard tools for computing the Euler
characteristic of finitely generated modules
over polynomial rings, the problem of computing
the curvature can be considered solved in these cases.
The problem of computing the curvature of ungraded
Hilbert modules remains open.
NOTE: A more detailed description of these results
has been published in Proc. Nat. Acad. Sci. USA,
vol. 96, pp. 11096--11099, September 1999, under the
title "The curvature invariant of a Hilbert module
over C[z_1,...,z_d]", and is available if pdf form
from the web site http://www.pnas.org/