Lifting of algebraic cycles and associated homomorphisms in intersection homology. (Relèvement de cycles algébriques et homomorphismes associés en homologie d’intersection.) (French) Zbl 0827.14012

The theory of intersection homology introduced by Goresky and MacPherson has had a profound influence on many areas of mathematics in the last decade and a half. Intersection homology has many advantages over other sorts of homology, but suffers from the disadvantage that it is not functorial in a natural sense. This paper provides an important partial remedy to this. The authors show that if \(f : X \to Y\) is a morphism of complex algebraic varieties of pure dimension, then there exist (not necessarily unique) homomorphisms \(\nu_f : IH_* (X) \to IH_* (Y)\) of intersection homology with rational coefficients and compact support for the middle perversity, called by the authors “associated homomorphisms”, such that the comparison homomorphisms \(IH_* (X) \to H_* (X)\) and \(IH_* (Y) \to H_* (Y)\) fit into a commutative square with \(\nu_f\) and \(f_* : H_* (X) \to H_* (Y)\). The authors deduce that homology classes of algebraic cycles admit liftings in intersection homology with rational coefficients having support arbitrarily close to the given cycles.
Reviewer: F.Kirwan (Oxford)


14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14C25 Algebraic cycles
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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