Hodge conjecture

The Hodge conjecture states that on a smooth projective complex variety, every rational cohomology class of type (p,p) is a rational linear combination of classes of algebraic cycles.

Hodge decomposition: A decomposition of the complex cohomology of a compact Kähler manifold into subspaces indexed by pairs of integers. It defines the type (p,p) condition central to the conjecture.
Hodge class: A rational cohomology class whose complexification lies in the (p,p) component of the Hodge decomposition. These are exactly the classes whose algebraicity the conjecture predicts.
Divisors: Codimension-one subvarieties, or formal combinations of them, on an algebraic variety. Their classes fall under the proven (1,1) case.
Integral Hodge conjecture: The claim that every integral Hodge class is an integral combination of algebraic cycle classes. It strengthens rational coefficients to integral ones and fails in general.
William Vallance Douglas Hodge: A Scottish mathematician whose work on harmonic integrals helped establish modern Hodge theory. His theory supplied the decomposition that motivates the conjecture.
Algebraic cycle: A finite formal integer combination of irreducible algebraic subvarieties of a fixed dimension. Its cohomology class is the kind of class the conjecture seeks to characterize.
Chow group: The group of algebraic cycles modulo rational equivalence. Cycle classes begin in these groups before mapping into cohomology.
Abelian variety: A projective algebraic group that is also a smooth connected projective variety. Products and special classes on abelian varieties provide prominent test cases.
Generalized Hodge conjecture: A conjectural refinement describing coniveau filtrations on cohomology through supports on algebraic subvarieties. It extends the question from individual Hodge classes to broader cohomological filtrations.
Lefschetz theorem on (1,1) classes: The theorem that integral cohomology classes of type (1,1) on a smooth projective complex variety are divisor classes. This known result is the conjecture's codimension-one precedent.

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