Filtrations on algebraic cycles and homology. (English) Zbl 0854.14006

Friedlander, Eric M.

Ann. Sci. Éc. Norm. Supér. (4) 28, No. 3, 317-343 (1995).

Summary: As shown by the author and B. Mazur [Mem. Am. Math. Soc. 529 (1994; Zbl 0841.14019)] Lawson homology theory determines natural filtrations on algebraic equivalence classes of algebraic cycles and on the singular integral homology groups of complex projective varieties. In this paper, the filtration on cycles is identified in terms of the images under correspondences of cycles homologically equivalent to zero. This is closely related to a filtration recently introduced by M. Nori. The author and B. Mazur conjectured that the filtration on (rational) homology groups was equal to the “geometric (or level) filtration” introduced by A. Grothendieck. It is shown here that this conjecture is implied by the validity of Grothendieck’s conjecture B. Both filtrations can be interpreted in terms of a spectral sequence whose various terms have a motivic nature.
The two central constructions of the paper are the \(s\)-map (introduced by the author and B. Mazur) and the graph mapping. Various equivalent descriptions of the \(s\)-map are presented and some of its basic properties are verified. The graph mapping is an elementary construction on cycle spaces which enables one to extend classical constructions involving correspondences to singular varieties.

MSC:

14C25	Algebraic cycles
14F99	(Co)homology theory in algebraic geometry

Keywords:

Lawson homology; algebraic equivalence classes of algebraic cycles; singular integral homology; graph mapping; \(s\)-map

Citations:

Zbl 0841.14019

References:

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