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Concept
Standard conjectures on algebraic cycles
Summary
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 that is . 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 H. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety
H ∗(X) → H ∗(X)
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.
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
W = H ∩ X,
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
L : H i(X) → H i+2(X),
which is defined by intersecting cohomology classes with W, gives an isomorphism
Ln−i : H i(X) → H 2n−i(X).
Now, for i ≤ n define:
Λ = (Ln−i+2)−1 ∘ L ∘ (Ln−i) : H i(X) → H i−2(X)
Λ = (Ln−i) ∘ L ∘ (Ln−i+2)−1 : H 2n−i+2(X) → H 2n−i(X)
The conjecture states that the Lefschetz operator (Λ) is induced by an algebraic cycle.
It is conjectured that the projectors
H ∗(X) ↠ Hi(X) ↣ H ∗(X)
are algebraic, i.
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