Standard Conjectures On Algebraic Cycles - 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 PN containing the given smooth projective variety X, the Lefschetz operator

L : Hi(X) → Hi+2,

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

Ln-i: Hi(X) → H2n-i(X) (i ≦ n = dim X).

Define

Λ : Hi(X) → Hi-2(X) for 'i ≦ n

be the composition

(Ln-i+2)-1 (Ln-i)

and

Λ : H2n-i+2(X) → H2n-i(X)

by

(Ln-i) (Ln-i+2)-1.

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

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