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

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 ${\pi }^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(X\mathrm{/}k)$, 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 ${\pi }^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(X\mathrm{/}k)$ 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.