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High Quality Content by WIKIPEDIA articles! The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, so there is a US$1,000,000 prize for proving the Hodge conjecture.