Hodge Conjecture - Statement of The Hodge Conjecture

Statement of The Hodge Conjecture

Let:

We call this the group of Hodge classes of degree 2k on X.

The modern statement of the Hodge conjecture is:

Hodge conjecture. Let X be a projective complex manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space carries a Kähler metric, the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogenous polynomials.