The Hodge conjecture asserts that, for smooth projective complex algebraic varieties, every Hodge cycle is a rational linear combination of cohomology classes of algebraic cycles.
Hodge Conjecture
See also
Algebraic Cycle, Algebraic Variety, Hodge Cycle, Millennium Prize ProblemsExplore with Wolfram|Alpha
References
Clay Mathematics Institute. "Hodge Conjecture." Accessed 11 Sep 2026. https://www.claymath.org/millennium/hodge-conjecture/.Deligne, P. "The Hodge Conjecture." https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf.Grothendieck, A. "Hodge's General Conjecture Is False for Trivial Reasons." Topology 8, 299-303, 1969.Hodge, W. V. D. "The Topological Invariants of Algebraic Varieties." Proc. Internat. Congress Math., Cambridge, Mass., 1950, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 182-192, 1952.Referenced on Wolfram|Alpha
Hodge ConjectureCite this as:
Weisstein, Eric W. "Hodge Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HodgeConjecture.html