Real intersection homology II: A local duality obstruction

Abstract

We prove that the intersection pairing on the real intersection homology of a real algebraic variety is not a dual pairing in general. A classical argument of Thom for manifolds, adapted to real intersection homology, shows that if this intersection pairing is nonsingular and X is the link of a point in a real algebraic variety, with the dimension of X even, then the intersection homology Euler characteristic Iχ(X) is even. Using an existence theorem of Akbulut and King [1] we show there is a singular algebraic surface X that is the link of a point in a 3-dimensional real algebraic variety, such that Iχ(X) is odd. Thus our definition of real intersection homology [6] does not have the key self-duality property suggested by Goresky and MacPherson ([3], p.227), though it does enjoy their small resolution property.