On the Hodge conjecture and a theorem of Nori
Let $Y$ be a projective complex variety endowed with an ample line bundle $L$. The Noether-Lefschetz locus $NL$ is the subset of $|L|$ parametrising smooth hypersurfaces of $Y$ which vanishing cohomology has a non-zero Hodge class. First, I will give an explicit asymptotic description of the components of small codimension of $NL$ for $L$ sufficiently ample and will show that for these components the Hodge class is in the image of the cycle map, as predicted by Hodge Conjecture. Next, I will explain a (partly conjectural) generalisation of this result and its links with Nori's connectivity theorem. I will also give an explicit asymptotic bound for Nori's theorem to hold and will give examples showing that this bound is optimal.