Hodge conjecture
The Hodge standard conjecture is not to be confused with the "Hodge conjecture" which states that for smooth projective varieties over , every rational -class is algebraic. The Hodge conjecture implies the Lefschetz and Künneth conjectures and conjecture D for varieties over fields of characteristic zero. The Tate conjecture implies Lefschetz, Künneth, and conjecture D for ℓ-adic cohomology over all fields.