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 an iterated universal extension, we show that Nori's fundamental group $π{1}^{N}(X,x)$ 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 $\cup: \mathrm{H}^{1}(X)\otimes\mathrm{H}^{1}(X) \rightarrow \mathrm{H}^{2}(X)$. This can be seen as an analogy of a classical fact on the de Rham fundamental group of compact Kähler manifolds. We also prove a homotopy exact sequence for Nori's fundamental group.