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hodge-conjecture
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It asks when topology on a complex shape comes from actual subvarieties cut out by polynomials. It is known for divisors, the Lefschetz (1,1) theorem, and fails for integer coefficients, as shown by Atiyah and Hirzebruch. It is the least elementary of the Millennium problems to state, and even the right formalization is work in progress.
Posed 1950, W. V. D. Hodge.
Prize
$1,000,000 Clay Millennium Prize