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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.08494 |
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| _version_ | 1866911671380869120 |
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| author | Yi, Xiaodong |
| author_facet | Yi, Xiaodong |
| contents | 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 $π_{1}^{N}(X,x)$ 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 analogue of a classical fact on the de Rham fundamental group of compact Kähler manifolds. As a byproduct, we also determine low degree group cohomology of the trivial representation of $π_{1}^{N}(X,x)$, notably in degree $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On iterated universal extensions and Nori's fundamental group of nilpotent bundles Yi, Xiaodong Algebraic Geometry 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 $π_{1}^{N}(X,x)$ 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 analogue of a classical fact on the de Rham fundamental group of compact Kähler manifolds. As a byproduct, we also determine low degree group cohomology of the trivial representation of $π_{1}^{N}(X,x)$, notably in degree $2$. |
| title | On iterated universal extensions and Nori's fundamental group of nilpotent bundles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.08494 |