Hodge Conjecture - Algebraicity of Hodge Loci

Algebraicity of Hodge Loci

The strongest evidence in favor of the Hodge conjecture is the algebraicity result of Cattani, Deligne & Kaplan (1995). Suppose that we vary the complex structure of X over a simply connected base. Then the topological cohomology of X does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations. Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.