On the Polynomial Cohomology of Affine Manifolds

It is well known that the real cohomology of a compact Riemannian manifold M is isomorphic to the algebra of its harmonic forms. When M is a fiat Riemannian manifold, i.e. a Euclidean manifold, a differential form is harmonic if and only if it is a parallel differential form. In the local Euclidean coordinate systems on M, such a differential form has constant coefficients. Thus, for a Euclidean manifold, there is canonical isomorphism between its real cohomology and its algebra of parallel differential forms. One can also ask about differential forms on M which in local coordinates have polynomial coefficients. A parallel form is one which is expressed by polynomials of degree 0. Unlike parallel forms, however, a polynomial form is not in general closed. The inclusion of the complex of polynomial forms on M into the de Rham complex of M induces a mapi , :H*oly (M)~H*(M ) of the polynomial cohomology of M into the de Rham cohomology of M. For a compact Euclidean manifold, a polynomial differential form on M must be parallel, so we get nothing new. However the notion of a polynomial form is invariant under more than just (isometric) Euclidean coordinate changes; rather the notion of polynomial form is invariant under affine coordinate changes. Hence if M is an affine manifold (see [ F G H ] for the precise definition), there is a well-defined complex ~/:oly(M) of polynomial forms, which is naturally a subcomplex of the C a de Rham complex d* (M) . The inclusion i: ~*o ly (M)cd*(M) induces a map i* on cohomology. Several properties of the map