Asymptotic bounds for Nori’s connectivity theorem

Let $Y$ be a smooth complex projective variety. I study the cohomology of smooth families of hypersurfaces $\mathcal {X}\to B$ for $B\subset \mathbb {P}\mathrm {H}^0(Y,\mathcal {O}(d))$ a codimension $c$ subvariety. I give an asymptotically optimal bound on $c$ and $k$ when $d\to \infty$ for the space $\mathrm {H}^k(Y\times B,\mathcal {X},\mathbb {Q})$ to vanish, thus extending the validity of the Lefschetz Hyperplane Section Theorem and Nori’s Connectivity Theorem (1993). Next, I construct in the limit case explicit families of higher Chow groups whose class does not vanish in $\mathrm {H}^k(Y\times B,\mathcal {X},\mathbb {Q})$. Some of them are indecomposable. This suggests that in the limit case the space $\mathrm {H}^k(Y\times B,\mathcal {X},\mathbb {Q})$ should be spanned by higher Chow groups, containing Nori’s and Otwinowska’s results as special cases.