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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

##### MSC:
 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)
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