Lectures on Hodge theory and algebraic cycles. (English) Zbl 1346.14025
These notes are lectures, a sequel to the author’s [Bol. Soc. Mat. Mex., III. Ser. 7, No. 2, 137–192 (2001; Zbl 1092.14013)]. The theme is precisely described by the keywords: Chow group; Hodge theory; algebraic cycle; regulator; Deligne cohomology; Beilinson-Hodge conjecture; Abel-Jacobi map; Bloch-Beilinson filtration. We have here a terse report of the past and present status of that fascinating part af algebraic geometry which deals with the Abel-Jacobi map as a tool for understanding the structure of algebraic cycles. It is quite useful, also because the author gives an interesting account of his and his collaborators’ recent work. Moreover we find a discussion of several conjectures and hints of the proofs which Lewis found to solve some of them. The bulk of the exposition centers on the cycle class maps, called either regulators or the Abel-Jacobi maps, which Bloch and Beilinson had constructed. The goal here is to describe them in an explicit way and then to apply the found formulas concretely. A brilliant student could profit greatly by this manual’s perusal, still he should be adviced that the discussion of ideas and proofs is necessarily sketchy and further understanding requires consultation of the sources.
Reviewer: Alberto Collino (Verzuolo)
MSC:
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 14C25 | Algebraic cycles |
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
Keywords:
Chow group; Hodge theory; algebraic cycle; regulator; Deligne cohomology; Beilinson-Hodge conjecture; Abel-Jacobi map; Bloch-Beilinson filtrationCitations:
Zbl 1092.14013References:
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