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.
Read more about this topic: Standard Conjectures On Algebraic Cycles
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