On iterated universal extensions and Nori's fundamental group of nilpotent bundles

Abstract

Let $k$ be a field of characteristic $0$, $X$ be a geometrically connected, smooth and proper variety over $k$ and $x\in X(k)$ be a base point. Using the notion of iterated universal extensions, we show that Nori's fundamental group ${\pi }_1^N(X,x)$ of nilpotent bundles is uniquely determined by the coherent cohomology groups ${\mathrm{H}}^i(X)={\mathrm{H}}^i(X,{\mathcal{O}}_X)$, $i=1,2$, and the cup product $\cup :{\mathrm{H}}^1(X)\otimes {\mathrm{H}}^1(X)\to {\mathrm{H}}^2(X)$. This can be seen as an analogue of a classical fact on the de Rham fundamental group of compact Kähler manifolds. As a byproduct, we also determine low degree group cohomology of the trivial representation of ${\pi }_1^N(X,x)$, notably in degree $2$.

The new version is based on the report from the referee, whom the author would like to thank heartily. Compared to the previous version, the discussion on the homotopy exact sequence is removed. Instead, we add a new section where we compute certain group cohomology in low degree. We also add some references