Standard conjectures on algebraic cycles

In mathematics, the standard conjectures about algebraic cycles is a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. The original application envisaged by Grothendieck was to prove that his construction of pure motives give an abelian category, that is semisimple. Moreover, as he pointed out, the standard conjectures also imply all Weil conjectures, including the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968).

The classical formulations of the standard conjectures involve a fixed Weil cohomology theory "H". All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on cohomology

:H*(X) → H*(X)

induced by an algebraic cycle on the product "X" × "X" via the "cycle class map "(which is part of the structure of a Weil cohomology theory).

Lefschetz type standard conjecture

"Lefschetz type standard conjecture", also called conjecture B: One of the axioms of a Weil theory is the so-called "hard Lefschetz theorem" (or axiom): for a fixed smooth hyperplane section

:"W" = "H" ∩ "X",

for "H" some hyperplane in the ambient projective space ℙ^N containing the given smooth projective variety "X", the Lefschetz operator

:"L" : "H^i"("X") → "H"^"i"+2,

which is defined by intersecting cohomology classes with "W" gives an isomorphism

:"L^n-i: Hⁱ(X) → H^2n-i(X)" ("i" ≦ "n = dim X").

Define

:"Λ : Hⁱ(X)" → "H^i-2(X)" for 'i" ≦ "n"

be the composition

:"(L^n-i+2)^-1 $circ\; L\; circ$ (L^n-i)^-1"

and

:"Λ : H^2n-i+2(X)" → "H^2n-i(X)"

by

:"(L^n-i) $circ\; L\; circ$ (L^n-i+2)^{-1 -1}".

The Lefschetz conjecture states that the "Lefschetz operator" "Λ" is induced by an algebraic cycle.

Künneth type standard conjecture

The Lefschetz conjecture implies the "Künneth type standard conjecture", also called conjecture C: It is conjectured that the projectors "H"^∗("X") ↠ "H"ⁱ("X") ↣ "H"^∗("X") are algebraic, i.e. induced by a cycle "πⁱ" ⊂ "X × X" with rational coefficients. This implies that every pure motive "M" decomposes in graded pieces of pure weights (see motives). The conjecture is known to hold for curves, surfaces and abelian varieties.

Conjecture "D" (numerical equivalence vs. homological equivalence)

"Conjecture D" states that numerical equivalence and homological equivalence agree. (In particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture.

Hodge conjecture

The "Hodge conjecture" is modelled on the Hodge index theorem states the positive definiteness of the cup product pairing on primitive algebraic cohomology classes or, equivalently, that every Hodge class is algebraic. If it holds, then the Lefschetz conjecture implies Conjecture "D". If the base field is of characteristic zero then the Hodge conjecture implies all the other standard conjectures (for a classical Weil cohomology theory, i.e. Betti, ℓ-adic, de Rham or crystalline cohomology).

References

*A. Grothendieck, [http://www.math.jussieu.fr/~leila/grothendieckcircle/StandardConjs.pdf "Standard Conjectures on Algebraic Cycles"] , Algebraic Geometry, Bombay Colloquium 1968, (OUP 1969) pp.193-199
* | year=1968 | chapter=Algebraic cycles and the Weil conjectures | pages=359–386
*S. L. Kleiman, "The standard conjectures". Motives (1994), pp. 3-20, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc.