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

TL;DR

This paper shows that Nori's fundamental group of nilpotent bundles for certain varieties is uniquely determined by specific cohomology groups and cup product, extending classical results on de Rham fundamental groups.

Contribution

It introduces the use of iterated universal extensions to characterize Nori's fundamental group via cohomology, providing a new perspective and explicit computations.

Findings

Nori's fundamental group is determined by H^1, H^2, and the cup product.

The approach extends classical results on de Rham fundamental groups.

Low degree group cohomology of the trivial representation is explicitly computed.

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 $\cup: \mathrm{H}^{1}(X)\otimes\mathrm{H}^{1}(X) \rightarrow \mathrm{H}^{2}(X)$. This can be seen as an analogue of a classical fact on the de Rham fundamental group of compact K\"ahler manifolds. As a byproduct, we also determine low degree group cohomology of the trivial representation of $\pi_{1}^{N}(X,x)$, notably in degree $2$.