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

Barthel, G.; Brasselet, J.-P.; Fieseler, K.-H.; Gabber, O.; Kaup, L.

Ann. Math. (2) 141, No. 1, 147-179 (1995).

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)

MSC:

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)

Keywords:

intersection homology; algebraic cycles; liftings