A morphism of intersection homology induced by an algebraic map

Abstract

Let f : X → Y be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups f∗ : IH∗(Y ) → IH∗(X) compatible with the induced homomorphism on cohomology. The crucial point in the argument is reduction to the finite characteristic. We give an alternative and short proof of the existence of a homomorphism f∗. Our construction is an easy application of the Decomposition Theorem. Let X be an algebraic variety, IH∗(X) = H∗(X ; ICX) its rational intersection homology group with respect to the middle perversity and ICX the intersection homology sheaf which is an object of derived category of sheaves over X [GM1]. We have the homomorphism ωX : H∗(X ; Q) −−→ IH∗(X) induced by the canonical morphism of the sheaves ωX : QX −−→ ICX . Let f : X −→ Y be a map of algebraic varieties. It induces a homomorphism of the cohomology groups. The natural question arises: Does there exist an induced homomorphism for intersection homology compatible with f∗ ? IH∗(Y ) ? −−→ IH∗(X) xωY xωX H∗(Y ; Q) f ∗ −−→ H∗(X ; Q) . The answer is positive. For topological reasons the map in question exists for normally nonsingular maps [GM1, §5.4.3] and for placid maps [GM3, §4]. The authors of [BBFGK] proved the following: Theorem 1. Let f : X −→ Y be an algebraic map of algebraic varieties. Then there exists a morphism λf : ICY −→ Rf∗ICX such that the following diagram with the canonical morphisms commutes: ICY λf −−→ Rf∗ICX xωY xRf∗(ωX) QY αf −−→ Rf∗QX . Received by the editors February 24, 1998. 1991 Mathematics Subject Classification. Primary 14F32, 32S60; Secondary 14B05, 14C25.