The Hodge Standard Conjecture
The Hodge standard conjecture is modelled on the Hodge index theorem. It states the positive definiteness 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 only for surfaces.
The Hodge standard conjecture is not to be confused with the Hodge conjecture which states that for smooth projective varieties over C, every rational (p,p)-class is algebraic. The Hodge conjecture implies the Lefschetz conjecture and conjecture D for varieties over fields of characteristic zero. Likewise for fields of finite characteristic the Tate conjectures in ℓ-adic cohomology imply the Lefschetz conjecture.
Read more about this topic: Standard Conjectures On Algebraic Cycles
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