The problem
On a smooth projective complex variety, every Hodge class (rational (p,p)-cohomology class) is a rational linear combination of classes of complex-algebraic subvarieties.
The Hodge Conjecture
openHistory & significance
Posed in Hodge's 1950 ICM address, reshaped by Grothendieck into its modern form. Known for divisors (Lefschetz (1,1)) and some abelian varieties and complete intersections; unknown already for general fourfolds. The conjecture anchors the bridge between transcendental and algebraic cycles that motives were invented to explain.
Still open.
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