  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).
