Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci

Abstract

We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensional linear projective space. In the particular case of surfaces in the projective space of dimension three, this is a theorem of Green and Voisin. In this case our classification under a computational hypothesis on IVHS implies a weaker version of the Harris-Voisin conjecture which says that the set of special components of the Noether-Lefschetz locus is not Zariski dense in the parameter space.