In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely 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 . The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.
The classical formulations of the standard conjectures involve a fixed Weil cohomology theory . All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety
induced by an algebraic cycle with rational coefficients on the product via the cycle class map, which is part of the structure of a Weil cohomology theory.
Conjecture A is equivalent to Conjecture B (see, p. 196), and so is not listed.
One of the axioms of a Weil theory is the so-called hard Lefschetz theorem (or axiom):
Begin with a fixed smooth hyperplane section
,
where is a given smooth projective variety in the ambient projective space and is a hyperplane. Then for, the Lefschetz operator
,
which is defined by intersecting cohomology classes with, gives an isomorphism
.
Now, for define:
The conjecture states that the Lefschetz operator is induced by an algebraic cycle.
It is conjectured that the projectors
are algebraic, i.e. induced by a cycle with rational coefficients. This implies that the motive of any smooth projective variety (and more generally, every pure motive) decomposes as
h(X)=
| 2dim(X) | |
| oplus | |
| i=0 |
hi(X).
The motives
h0(X)
h2
proved the Künneth decomposition for abelian varieties A. refined this result by exhibiting a functorial Künneth decomposition of the Chow motive of A such that the n-multiplication on the abelian variety acts as
ni
hi(A)
Conjecture D states that numerical and homological equivalence agree. (It implies in particular the latter does not depend on the choice of the Weil cohomology theory). This conjecture implies the Lefschetz conjecture. If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent.
This conjecture was shown by Lieberman for varieties of dimension at most 4, and for abelian varieties.
The Hodge standard conjecture is modelled on the Hodge index theorem. It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes. If it holds, then the Lefschetz conjecture implies Conjecture D. In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic the Hodge standard conjecture is known for surfaces and for abelian varieties of dimension 4 .
The Hodge standard conjecture is not to be confused with the Hodge conjecture which states that for smooth projective varieties over, every rational -class is algebraic. The Hodge conjecture implies the Lefschetz and Künneth conjectures and conjecture D for varieties over fields of characteristic zero. The Tate conjecture implies Lefschetz, Künneth, and conjecture D for ℓ-adic cohomology over all fields.
For two algebraic varieties X and Y, has introduced a condition that Y is motivated by X. The precise condition is that the motive of Y is (in André's category of motives) expressible starting from the motive of X by means of sums, summands, and products. For example, Y is motivated if there is a surjective morphism
Xn\toY
has shown that the (conjectural) existence of the so-called motivic t-structure on the triangulated category of motives implies the Lefschetz and Künneth standard conjectures B and C.