Filtrations on the homology of algebraic varieties / Eric M. Friedlander, Barry Mazur

This work provides a detailed exposition of several topics related to Lawson homology, mixed Hodge structures and algebraic cycles; moreover a filtration on Betti homology is introduced as the span of fundamental classes given by Lawson homology classes and it is herein proved to be finer than the niveau filtration. In fact, for Lawson homology L j H n (X) of a (smooth) projective variety X over the complex numbers, we have that L 0 H n (X) is naturally isomorphic to singular homology H n (X); furthermore we have operations L j H n (X)→L j-1 H n (X). Iterating these operations we obtain a canonical homomorphism L j H n (X)→H n (X): its image gives us the claimed filtration T j H n (X) called topological. Grothendieck niveau filtration G j H n (X) is given by homology classes in H n (X) which are supported on algebraic subvarieties of X of dimension ≤n-j. It is then proved that T j H n (X) is contained in G j H n (X). The proof involves the use of a Hopf algebra with values in the abelian category of limits of mixed Hodge structures (rational Lawson homology is canonically isomorphic to a direct limit of homotopy groups of Chow monoids). An interpretation of the topological filtration in terms of correspondences is given in order to compare it with the niveau filtration. Several questions and speculations are proposed to the reader. Finally there are nice appendixes on mixed Hodge structures, cycle classes, algebraic suspension, the Dold-Thom theorem using trace maps and, notably, an unpublished paper by D. Quillen concerning the group completion of a simplicial monoid. (Zentralblatt)

Bibliogr. p. 107-109