Abelian varieties are de Rham $K(π,1)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914288610836480 |
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| author | Bao, Vo Quoc Nguyen, Quang-Khai |
| author_facet | Bao, Vo Quoc Nguyen, Quang-Khai |
| contents | Motivated by the work of Esnault-Hai, one has the notion of de Rham $K(π,1)$ 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 $π^{\mathrm{diff}}(X/k)$, which comes from the Tannakian duality of the category of coherent integrable connections on $X$. Using the formalism of $δ$-functors, one can define natural morphisms between the group-scheme cohomology of $π^{\mathrm{diff}}(X/k)$ and the de Rham cohomology of $X$. One says that $X$ with $x\in X(k)$ is de Rham $K(π,1)$ if such morphisms are all isomorphisms. In this article, we first prove that abelian varieties in characteristic $0$ are de Rham $K(π,1)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01595 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Abelian varieties are de Rham $K(π,1)$ Bao, Vo Quoc Nguyen, Quang-Khai Algebraic Geometry 14F40, 14F43, 14L15, 18G15, 18G40, 18M25 Motivated by the work of Esnault-Hai, one has the notion of de Rham $K(π,1)$ 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 $π^{\mathrm{diff}}(X/k)$, which comes from the Tannakian duality of the category of coherent integrable connections on $X$. Using the formalism of $δ$-functors, one can define natural morphisms between the group-scheme cohomology of $π^{\mathrm{diff}}(X/k)$ and the de Rham cohomology of $X$. One says that $X$ with $x\in X(k)$ is de Rham $K(π,1)$ if such morphisms are all isomorphisms. In this article, we first prove that abelian varieties in characteristic $0$ are de Rham $K(π,1)$. 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. |
| title | Abelian varieties are de Rham $K(π,1)$ |
| topic | Algebraic Geometry 14F40, 14F43, 14L15, 18G15, 18G40, 18M25 |
| url | https://arxiv.org/abs/2601.01595 |