Hodge conjecture
By the strong and weak Lefschetz theorem, the only non-trivial part of the Hodge conjecture for hypersurfaces is the degree "m" part (i.e., the middle cohomology) of a 2"m"-dimensional hypersurface formula_42. If the degree "d" is 2, i.e., "X" is a quadric, the Hodge conjecture holds for all "m". For formula_43, i.e., fourfolds, the Hodge conjecture is known for formula_44.