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The Hodge cohomology and cubic equivalences. (English) Zbl 0574.14005
Let V be a smooth projective variety and f a family of effective cycles of dimension r on V parametrised by a smooth variety S. Fix a positive integer \(\ell\) and consider a cohomology class \(\omega\) in \(H^ r(V,\Omega^{r+\ell})\). In analogy with the case of 0-cycles, studied by Mumford and Roitman, one defines and studies the associated form \(f^{\#}\omega\) on S. If g is another family such that f(s) and g(s) are rationally equivalent for every s, then \(f^{\#}\omega =g^{\#}\omega\). The main results concern the \(\ell\)-cubic equivalence relation in sense of Samuel: \(f^{\#}\omega\) cannot distinguish the \(\ell '\)-cubic equivalence relation for \(\ell '>\ell\), but happens to distinguish the \(\ell\)-cubic one. The associated Chow groups are also studied.
Reviewer: C.Bănică

14C15 (Equivariant) Chow groups and rings; motives
14C05 Parametrization (Chow and Hilbert schemes)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
Full Text: DOI
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