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standard conjectures on algebraic cycles

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

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 (Grothendieck - 68), 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). 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). (Grothendieck - 68)

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.

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
  • 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)
  • wikipedia standard conjectures on algebraic cycles

For the Beilinson conjectures, see the references there.

Revised on October 24, 2014 20:07:09 by Adeel Khan (77.9.206.161)