×

Some results about Abel-Jacobi mappings. (English) Zbl 0575.14007

Topics in transcendental algebraic geometry, Ann. Math. Stud. 106, 289-304 (1984).
[For the entire collection see Zbl 0528.00004.]
The article is centered around the Hodge-Grothendieck conjecture concerning algebraic cycles. For an algebraic threefold V, the conjecture says: let \(H\subset H^ 3(V)\) be a sub-Hodge structure defined over \({\mathbb{Q}}\) and perpendicular to \(H^{3,0}(V)\). Then the natural projection E of \(H_{{\mathbb{C}}}\) viewed as a subspace of \(H^ 3(V,{\mathbb{C}})^*\) lies in the image of the Abel-Jacobi mapping. That is E is parametrized by algebraic 1-cycles.
This article is a review of the most interesting results obtained in this area, with some ideas on the proofs and possible extensions. - The paper ends with an heuristic, very convincing argument for proving that homological equivalence modulo algebraic equivalence is not finitely generated. [A complete proof of this important result is to be found in the author’s paper in Publ. Math., Inst. Haut. Étud. Sci. 58, 231-250 81983; Zbl 0529.14002)].
Reviewer: F.Gherardelli

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C15 (Equivariant) Chow groups and rings; motives