Hodge Conjecture
Predicts which cohomology classes on projective varieties come from algebraic cycles.
The Hodge Conjecture asserts that on a smooth complex projective variety X, every rational cohomology class of type (p,p) -- that is, every class in the intersection of H^{2p}(X, Q) with H^{p,p}(X) -- is a rational linear combination of classes of algebraic subvarieties. It is a deep bridge connecting algebraic geometry, topology, and complex analysis. The conjecture was presented by W. V. D. Hodge at the 1950 International Congress of Mathematicians and is known to hold for divisors (the Lefschetz (1,1)-theorem, proved in 1924) and in certain low-dimensional cases. Voisin showed in 2002 that the generalization to compact Kahler manifolds is false, confirming the conjecture is intrinsically about projective algebraic geometry.
Formula
Every Hodge class on a smooth projective variety should be a rational linear combination of classes of algebraic subvarieties.
The cohomology of a compact Kähler manifold splits into (p,q)-types — the Hodge structure underlying the conjecture.
For p = 1 the Hodge conjecture is the Lefschetz (1,1)-theorem: every Hodge class of type (1,1) is algebraic.
Summary
The Hodge Conjecture asserts that on a smooth complex projective variety X, every rational cohomology class of type (p,p) -- that is, every class in the intersection of H^{2p}(X, Q) with H^{p,p}(X) -- is a rational linear combination of classes of algebraic subvarieties. It is a deep bridge connecting algebraic geometry, topology, and complex analysis. The conjecture was presented by W. V. D. Hodge at the 1950 International Congress of Mathematicians and is known to hold for divisors (the Lefschetz (1,1)-theorem, proved in 1924) and in certain low-dimensional cases. Voisin showed in 2002 that the generalization to compact Kahler manifolds is false, confirming the conjecture is intrinsically about projective algebraic geometry.
