nLab
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, 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

For the Beilinson conjectures, see the references there.

Revised on August 24, 2015 12:41:57 by M. Carmona? (190.128.61.5)