Speaker
Description
Let $X$ be a smooth and projective variety of dimension $n+1$ and let $f\colon \mathcal{Y} \to S$ be the universal family of smooth hypersurfaces in $X$ of a fixed degree. Assuming that the degree is sufficiently large, Nori proved that the cohomology of the base change $\mathcal{Y}\times_S T$ of $\mathcal{Y}$ along a smooth morphism $T\to S$ coincides with the cohomology of $X\times T$ up to degree $2n-1$. In particular, this gives a simple way to compute the cohomology of the local system $(R^n f_*\mathbb{Q})_{\mathsf{prim}}$ up to degree $n-1$, and after base change by any smooth morphism.
We propose a version of Nori's theorem for the self product $\mathcal{Y}\times_S \mathcal{Y}$ of the universal family. This will yield information on the higher endomorphisms of the local system $(R^nf_*\mathbb{Q})_{\mathsf{prim}}$. We hope to use these endomorphisms to rule out the existence of positive degree operations on rational Betti cohomology and hence proving that the motivic Galois group is classical.
