nLab Standard Conjectures on Algebraic Cycles

Idea

What are called the standard conjectures on algebraic cycles are several conjectures brought up by Grothendieck, concerned with the relation between algebraic cycles and Weil cohomology theories.

“The first (Lefschetz standard conjecture) is an existence assertion for algebraic cycles, the second (Hodge standard conjecture) is a statement of positivity, generalising Weil’s well-known positivity theorem in the theory of abelian varieties” (Grothendieck - 68).

“But in the category of pure motives, from the start one dealt only with algebraic cycles, represented by correspondences, and it was intuitively not at all clear how on earth they could convey information about transcendental cycles. Indeed, the main function of the ”Standard Conjectures“ was to serve as a convenient bridge from algebraic to transcendental. Everything that one could prove without them was indeed ”plus ou moins trivial“ – until people started treating correspondences themselves using sophisticated homological algebra (partly generated by the development of ́étale cohomology and Grothendieck–Verdier’s introduction of derived and triangulated categories). (Y. Manin - 2014).

They were also followed by the Beilinson conjectures“.

Conclusions

The proof of the two standard conjectures would yield results going considerably further than Weil’s conjectures. They would form the basis of the so-called “theory of motives” which is a systematic theory of “arithmetic properties” of algebraic variety(ies), as embodied in their groups of classes of cycles for numerical equivalence. We have at present only a very small part of this theory in dimension one, as contained in the theory of abelian variety(ies). Alongside the problem of resolution of singularities, the proof of the standard conjectures seems to me to be the most urgent task in algebraic geometry. (Grothendieck - 68)

References

• A. Grothendieck, Standard Conjectures on Algebraic Cycles, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, pp. 193–199, (pdf)
• Steven Kleiman, Algebraic cycles and the Weil conjectures, in: Dix exposés sur la cohomologie des schémas, North-Holland, pp. 359–386, 1968, MR0292838; The standard conjectures, Motives (Seattle, WA, 1991), Proc. of Symposia in Pure Math. 55, American Mathematical Society, pp. 3–20, 1994, MR1265519
• Yuri Manin, Forgotten Motives: the Varieties of Scientific Experience (arXiv:1402.2155)
• Alexander Beilinson, Remarks on Grothendieck’s standard conjectures (arXiv:math/1006.116)
• James Milne, Polarizations and Grothendieck’s standard conjectures (arXiv:math/0103175)
• Yves Andre, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses 17, Paris: Société Mathématique de France (2204) (pdf)
• Mathoverflow, Progress on the standard conjectures on algebraic cycles (pdf)
• Voevodsky, From motives to motivic homotopy types, Visions in mathematics, Tel Aviv, August-September 1999 (pdf)
• K. Voelkel?, Seminar on motives, Standard conjectures (pdf)
• Secret blogging seminar, The Weil conjectures: the approach via the standard conjectures (pdf)
• wikipedia standard conjectures on algebraic cycles
• Serre, Analogues Kähleriens de certaines conjectures de Weil (extrait d’une lettre a A. Weil, 9 Nov. 1959). (pdf)

For the Beilinson conjectures, see the references there.