On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909997361790976 |
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| author | Shimoji, Taito |
| author_facet | Shimoji, Taito |
| contents | Let $X$ be a smooth quasi-projective variety. Assume that the (topological) fundamental group $π_1(X, x)$ is torsion-free nilpotent. We show that if the first Betti number $b_1(X) \le 3$, then $π_1(X, x)$ is isomorphic to either $\mathbb{Z}^n$ for $n = 1, 2, 3$, a lattice in the Heisenberg group $H_3(\mathbb{R})$ or $\mathbb{R} \times H_3(\mathbb{R})$. Moreover, we prove that $π_1(X, x)$ is abelian or $2$-step nilpotent if its rank is less than or equal to seven. More precisely, we determine the real nilpotent Lie groups in which torsion-free nilpotent fundamental groups can be embedded as lattices for ranks up to six and seven, respectively. Our main results are a partial positive answer to a question on nilpotent (quasi-)Kähler groups posed by Aguilar and Campana. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven Shimoji, Taito Algebraic Geometry Algebraic Topology Differential Geometry 14F35, 11F23, 55P62 Let $X$ be a smooth quasi-projective variety. Assume that the (topological) fundamental group $π_1(X, x)$ is torsion-free nilpotent. We show that if the first Betti number $b_1(X) \le 3$, then $π_1(X, x)$ is isomorphic to either $\mathbb{Z}^n$ for $n = 1, 2, 3$, a lattice in the Heisenberg group $H_3(\mathbb{R})$ or $\mathbb{R} \times H_3(\mathbb{R})$. Moreover, we prove that $π_1(X, x)$ is abelian or $2$-step nilpotent if its rank is less than or equal to seven. More precisely, we determine the real nilpotent Lie groups in which torsion-free nilpotent fundamental groups can be embedded as lattices for ranks up to six and seven, respectively. Our main results are a partial positive answer to a question on nilpotent (quasi-)Kähler groups posed by Aguilar and Campana. |
| title | On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven |
| topic | Algebraic Geometry Algebraic Topology Differential Geometry 14F35, 11F23, 55P62 |
| url | https://arxiv.org/abs/2510.09026 |