Real intersection homology

Abstract

Abstract. We present a definition of intersection homology for real algebraic varieties that is analogous to Goresky and MacPherson's original definition of intersection homology for complex varieties.Let X be a real algebraic variety. For certain stratifications S of X we define homology groups IH S k (X) with Z/2 coefficients that generalize the standard intersection homology groups [3] if all strata have even codimension. Whether there is a good analog of intersection homology for real algebraic varieties was stated as a problem by Goresky and MacPherson [6] (Problem 7, p. 227). They observed that if such a theory exists then it cannot be purely topological; indeed our groups are not homeomorphism invariants.We consider a class of algebraic stratifications introduced in [16] that have a natural general position property for semialgebraic subsets. After presenting the definition and properties of these stratifications S, we define the real intersection homology groups IH S k (X) and show that they are independent of the stratification. We prove that if X is nonsingular and pure dimensional then IH k (X) = H k (X; Z/2), classical homology with Z/2 coefficients. More generally, we prove that if X is irreducible and X admits a small resolution π : X → X then IH k (X) is canonically isomorphic to H k ( X; Z/2). Thus any two small resolution of X have the same homology.If X is not compact, we have two versions of real intersection homology: IH c k (X) with compact supports and IH cl k (X) with closed supports. We define an intersection pairing IH c k (X) × IH cl n−k (X) → Z/2, where n = dim X. We prove that if X has isolated singularities this pairing is nonsingular, so IH c k (X) ∼ = IH cl n−k (X) for all k ≥ 0. But our intersection pairing is singular for some real algebraic varieties X, so our groups fail to have the key selfduality property suggested by Goresky and MacPherson. We will present a counterexample in a subsequent paper.We benefitted from conversations more than a decade ago with Joost van Hamel, whose approach to real intersection homology [8] was quite different from ours. We dedicate this paper to his memory. Good algebraic stratificationsAn algebraic stratification of a real algebraic variety X is the stratification associated to a filtration of X by algebraic subvarietiessuch that for each j, Sing(X j ) ⊂ X j−1 , and either dim X j = j or X j = X j−1 . This filtration induces a decomposition X = S i , where the S i are the connected components of all the semialgebraic sets X j \ X j−1 . The sets S i , which are nonsingular semialgebraic subsets of X, 1

Real intersection homology II: A local duality obstruction

“…To simplify notation, we will identify a semialgebraic set with the chain it represents. The intersection homology groups are defined using allowable chains, which are represented by semialgebraic sets satisfying certain perversity conditions with respect to a good stratification (see [6]).…”

“…Suppose that a is represented by a compact allowable cycle A, and b is represented by a compact allowable cycle B, with A (stratified) transverse to B. (See [6]. This means that A has a semialgebraic stratification A, and B has a semialgebraic stratification B, such that A and B are substratified objects of the good stratification S|N , and for all strata S ∈ A, T ∈ B, with S and T contained in a stratum U ∈ S|N , we have that S and T are transverse in U .)…”

“…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.…”

Algebro-geometric equisingularity of Zariski

“…There is an open dense semialgebraic subset U ′ of U such that, for all t ∈ U ′ , (Z, A) is transverse to Ψ −1 t (W, B) in (P n , S). In a recent paper [39] Corollary 2.18 is used to define an intersection pairing for real intersection homology, an analog of intersection homology for real algebraic varieties.…”

Algebraic stratified general position and transversality

“…(In their initial paper [1], Goresky and MacPherson considered piecewise linear chains with respect to a triangulation.) In a subsequent paper [4] we will apply our transversality theorem to define an intersection pairing for real intersection homology, an analog of intersection homology for real algebraic varieties.…”