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.
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 H. 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 X × X via the cycle class map, which is part of the structure of a Weil cohomology theory.
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 X is a given smooth projective variety in the ambient projective space P N and H is a hyperplane.
Then for i ≤ n = dim(X), the Lefschetz operator which is defined by intersecting cohomology classes with W, gives an isomorphism Now, for i ≤ n define: The conjecture states that the Lefschetz operator (Λ) is induced by an algebraic cycle.
It is conjectured that the projectors are algebraic, i.e. induced by a cycle π i ⊂ X × X with rational coefficients.
Šermenev (1974) proved the Künneth decomposition for abelian varieties A. Deninger & Murre (1991) 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
de Cataldo & Migliorini (2002) proved the Künneth decomposition for the Hilbert scheme of points in a smooth surface.
It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes.
In positive characteristic the Hodge standard conjecture is known for surfaces (Grothendieck (1958)) and for abelian varieties of dimension 4 (Ancona (2020)).
[3] This fact can be applied to show, for example, the Lefschetz conjecture for the Hilbert scheme of points on an algebraic surface.