  We reformulate the integrality property of the Poincar\'{e} inner product in
the middle dimension, for an arbitrary Poincar\'{e} $\Q$-algebra, in classical
terms (discriminant and local invariants). When the algebra is 1-connected, we
show that this property is the only obstruction to realizing it by a closed
manifold, up to dimension 11. We reinterpret a result of Eisenbud and Levine on
finite map germs, relating the degree of the map germ to the signature of the
associated local ring, to answer a question of Halperin on artinian weighted
complete intersections.We analyse the homogeneous artinian complete
intersections over $\Q$ realized by closed manifolds of dimensions 4 and 8, and
their signatures.
