Abelian varieties are de Rham $K(\pi,1)$

TL;DR

This paper proves that abelian varieties over characteristic zero fields are de Rham $K(\pi,1)$ schemes, linking their differential fundamental groups to de Rham cohomology, and explores cohomology of their abelianized fundamental groups.

Contribution

It establishes that abelian varieties are de Rham $K(\pi,1)$ schemes and analyzes the cohomology of their differential fundamental groups via Albanese varieties.

Findings

Abelian varieties are de Rham $K(\pi,1)$ in characteristic zero.

Group-scheme cohomology of the abelianized differential fundamental group is studied.

Connections between differential fundamental groups and de Rham cohomology are clarified.

Abstract

Motivated by the work of Esnault-Hai, one has the notion of de Rham $K(\pi,1)$ schemes, defined as follows. Given a smooth proper geometrically connected scheme $X$ over a field $k$ of characteristic 0 and a base point $x \in X (k)$, one can define its differential fundamental group $\pi^{\mathrm{diff}}(X/k)$, which comes from the Tannakian duality of the category of coherent integrable connections on $X$. Using the formalism of $\delta$-functors, one can define natural morphisms between the group-scheme cohomology of $\pi^{\mathrm{diff}}(X/k)$ and the de Rham cohomology of $X$. One says that $X$ with $x\in X(k)$ is de Rham $K(\pi,1)$ if such morphisms are all isomorphisms. In this article, we first prove that abelian varieties in characteristic $0$ are de Rham $K(\pi,1)$. In the second part of the article, we study the group-scheme cohomology of the abelianization of the differential fundamental group of a smooth proper geometrically connected scheme via its Albanese variety.