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 g...