Algebraic cycles
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General Introduction
To a scheme one can associate its group of algebraic cycles, which is graded by codimension (or dimension). Algebraic cycle groups play a fundamental role in all kinds of cohomology theories in algebraic geometry. Almost every (ordinary?) cohomology theory admits a cycle map, i.e. a natural transformation from the algebraic cycles to the cohomology groups.
The theory of algebraic cycles is dominated by a series of unsolved conjectures, such as the Hodge conjecture, the Tate conjectures, Grothendieck's standard conjectures, etc. Some introductory surveys on these things can be found in the book of Andre on motives and in the Motives volume 1. An important textbook is Fulton's book on intersection theory, but other than that I am not aware of any introductory textbook on algebraic cycles.
See also Pure motives, Chow groups.
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Search results
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Online References
Jannsen: Equivalence relations on algebraic cycles (also a copy on my computer)
Very brief review of intersection theory: de Jong
Possibly of some interest: K0047, K0381
See Bloch and Ogus, especially the section on algebraic cycles.
Bourbaki exposes: 9,
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Paper References
Samuel: Relation d'equivalence en geom alg. In ICM 1960
Some Springer Lecture notes
Murre: Algebraic cycles and algebraic aspects of cohomology and K-theory. Voisin: Transcendental methods in the study of algebraic cycles
Bloch: Lectures on algebraic cycles. Duke Univ. Math series, No IV, 1980, or 1982.
Kleiman: Algebraic cycles and the Weil conjectures. In Dix exposes, 1968.
Kleiman: Chapter in the Motives volumes
Kleiman, Steven L. Finiteness theorems for algebraic cycles. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 445--449
Saito: Some remarks on the Hodge type conjectures (Motives vol)
André: Une introduction aux motifs
Saito: Motives and filtrations on Chow groups
Letter from Grothendieck to Serre on the standard conjectures, see page 232 in Groth-Serre.
Tate: Algebraic cycles and poles of zeta functions
MR1736872 (2001b:14002) The arithmetic and geometry of algebraic cycles. Proceedings of the CRM Summer School held in Banff, AB, June 7--19, 1998. Edited by B. Brent Gordon, James D. Lewis, Stefan Müller-Stach, Shuji Saito and Noriki Yui.
Algebraic cycles and motives. Vol. 1+2, London Math. Soc. Lecture Note Ser., 344, Cambridge Univ. Press, Cambridge, 2007
Laterveer: Intro to algebraic cycles and motives, background to the author's work on codim 2 cycles.
For transcendental methods and intermediate Jacobians: Griffiths lectures in LNM0185
Green and Griffiths: On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Google Books link
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Definition
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Properties
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Standard theorems
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Open Problems
Thomason: The classification of triangulated subcategories. In addition to classification (relating subcats to subgroups of the Grothendieck group) and relation to the Nilpotence theorem, this contains some speculation on the "right" definition of algebraic n-cycles, which should be an object in some triangulated subcategory of the category of perfect complexes. These cycles should give a good notion replacing the Chow ring in cases such as singular algebraic varieties and schemes flat and of finite type over Spec Z. The article also reviews basic notions in triangulated categories, such as thick subcategories, the Grothendieck group, and more.
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Perhaps one could check all articles under the classification code for algebraic cycles in MathSciNet
For examples of current research, check articles of Francois Charles, a young French super-star working on these things.
Algebraic cycles on degenerate fibers. Bloch, Gillet, Soulé
Tankeev: Very interesting article on conjectures on algebraic cycles. Check this.
Also worth checking: Other articles of Tankeev
Akhtar: Zero-cycles on varieties over finite fields (2004)
Spiess: Proof of the Tate conjecture for products of elliptic curves over finite fields (1999)
Lieberman: Numerical and homological equivalence of algebraic cycles on Hodge manifolds (1968)
Katz and Messing: Some remarks on the Tate conjecture over finite fields and its relations to other conjectures. Manuscript notes, 1991.
Schoen: On Hodge structures and non-representability of Chow groups
Jannsen: Motives, numerical equivalence, and semisimplicity
Levine: Techniques of localization...
Spencer Bloch, Cycles and biextensions (pp. 19--30) (1987, some proceedings)
Wayne Raskind, Algebraic $K$-theory, étale cohomology and torsion algebraic cycles (pp. 311--341);
R. W. Thomason, A finiteness condition equivalent to the Tate conjecture over $\bold F_q$ (pp. 385--392) (1987)
MR1403971 (98a:14016) Voisin, Claire(F-PARIS11) Variations of Hodge structure and algebraic cycles. (This MIGHT be a good survey)
MR1896474 (2003g:14007) del Angel, Pedro Luis(MEX-CIM); Müller-Stach, Stefan J.(D-ESSN) The transcendental part of the regulator map for $K\sb 1$ on a mirror family of $K3$-surfaces.
Geisser: Tate's conjecture, algebraic cycles, and rational K-theory in characteristic p
Green and Griffiths: Hodge-theoretic invariants for algebraic cycles
arXiv:1009.1434 Transcendence degree of zero-cycles and the structure of Chow motives from arXiv Front: math.AG by Sergey Gorchinskiy, Vladimir Guletskii We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Bloch's conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.
arXiv:0908.0626 Algebraic cycles on an abelian variety from arXiv Front: math.AG by Peter O'Sullivan It is shown that to every Q-linear cycle \bar\alpha modulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle \alpha modulo rational equivalence on A lying above \bar\alpha. The assignment \bar\alpha -> \alpha respects the algebraic operations and pullback and push forward along homomorphisms of abelian varieties.
arXiv:1003.0264 A cdh approach to zero-cycles on singular varieties from arXiv Front: math.AG by Amalendu Krishna We study the Chow group of zero-cycles on singular varieties using the cdh topology. We define the cdh versions of the zero-cycles and albanese maps. We prove results comparing these groups for a singular variety with the similar groups on the resolution of singularities. We use these to prove some results about the known Chow group of zero-cycles on surfaces and threefolds and some cases of arbitrary dimension.
math/0605165 A Finiteness theorem for zero-cycles over $p$-adic fields from arXiv Front: math.AG by Shuji Saito, Kanetomo Sato In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields.
In this final version, there are several corrections concerning mathematical symbols and reference to related known results.arXiv:0908.1927 Smash-nilpotent cycles on abelian 3-folds from arXiv Front: math.AG by Bruno Kahn, Ronnie Sebastian We show that homologically trivial algebraic cycles on a 3-dimensional abelian variety are smash-nilpotent.
arXiv:1207.7344 Smash nilpotent cycles on products of curves from arXiv Front: math.AG by Ronnie Sebastian Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove the conjecture for one dimensional cycles on an arbitrary product of curves. As a consequence we get that numerically trivial 1-cycles on an abelian variety are smash nilpotent.
arXiv:0911.3639 Relative Chow-Kuenneth decompositions for morphisms of threefolds from arXiv Front: math.AG by Stefan Mueller-Stach, Morihiko Saito We show that any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition. As a corollary we get sufficient conditions for threefolds to admit an absolute Chow-Kuenneth decomposition. In case the image of the morphism is a surface, this implies another proof of a theorem on the absolute Chow-Kuenneth decomposition for threefolds satisfying a certain condition, which was obtained by the first author with P. L. del Angel. In case the image is a curve, this improves in the threefold case a theorem obtained by the second author where the singularity of the morphism was assumed isolated and the condition on the general fiber was stronger.
arXiv:1101.3647 Notes on absolute Hodge classes from arXiv Front: math.AG by François Charles, Christian Schnell We survey the theory of absolute Hodge classes. The notes include a full proof of Deligne's theorem on absolute Hodge classes on abelian varieties as well as a discussion of other topics, such as the field of definition of Hodge loci and the Kuga-Satake construction.
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Other Information
[CDATA[Voevodsky: Nilpotence theorem for cycles algebraically equivalent to zero. Discusses smash nilpotence, the nilpotence conjectures, and various ideas related to mixed motives, theories of motivic type, the standard conjectures, and stuff about algebraic equivalence, for example mixed motives modulo alg equiv.
http://mathoverflow.net/questions/65377/dimension-of-chow-groups
http://mathoverflow.net/questions/11621/obstructions-to-descend-galois-invariant-cycles
http://mathoverflow.net/questions/11774/difference-between-equivalence-relations-on-algebraic-cycles
http://mathoverflow.net/questions/13985/why-are-cohomologically-trivial-cycles-abundant
http://mathoverflow.net/questions/13839/what-do-intermediate-jacobians-do
http://mathoverflow.net/questions/79376/reference-for-numerical-vs-homological-equivalence
http://mathoverflow.net/questions/15001/algebraic-equivalence-vs-numerical-equivalence-an-example
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