The status of the Hodge conjecture

[This is a guest post by Claire Voisin. This blog post was initially written in a different file format and converted using AI. — T.] Hodge classes can be defined on any compact complex manifold . They are rational Betti cohomology classes on of even degree (eg combinations with -coefficients of classes of oriented codimension closed submanifolds of ) which satisfy a delicate “Hodge condition” necessary for them to be combinations with -coefficients of classes of complex submanifolds (and more generally closed analytic spaces) of . To understand this condition, we need to pass to cohomology with complex coefficients, and represent complex cohomology classes as de Rham cohomology classes of closed forms. The Hodge condition is that the class should be representable by a closed form that in local holomorphic coordinates is written as . The Hodge conjecture states that a Hodge class on a smooth complex projective variety is “algebraic”, that is, is a combination with rational…