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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.06250 |
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| _version_ | 1866916796403023872 |
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| author | Rogov, Vasily |
| author_facet | Rogov, Vasily |
| contents | Let $X$ be a smooth complex quasi-projective variety and $Γ=π_1(X)$. Let $χ\colon Γ\to \mathbb{R}$ be an additive character. We prove that the ray $[χ]$ does not belong to the BNS set $Σ(Γ)$ if and only if it comes as a pullback along an algebraic fibration $f \colon X \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. We also prove that if $π_1(X)$ admits a solvable quotient which is not virtually nilpotent, there exists a finite étale cover $X_1 \to X$ and a fibration $f \colon X_1 \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. Both of these results were proved by Delzant in the case when $X$ is a compact Kähler manifold. We deduce that $Γ$ is virtually solvable if and only if it is virtually nilpotent, generalising the theorems of Delzant and Arapura-Nori. As a byproduct, we prove a version of Simpson's Lefschetz Theorem for the integral leaves of logarithmic $1$-forms that do not extend to any partial compactification. We give two applications of our results. First, we strengthen the recent theorem of Cadorel-Deng-Yamanoi on virtual nilpotency of fundamental groups of quasi-projective $h$-special and weakly special manifolds. Second, we prove the sharpness of Suciu's tropical bound for the fundamental groups of smooth quasi-projective varieties and answer a question of Suciu on the topology of hyperplane arrangements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Bieri-Neumann-Strebel sets of quasi-projective groups Rogov, Vasily Algebraic Geometry Group Theory Let $X$ be a smooth complex quasi-projective variety and $Γ=π_1(X)$. Let $χ\colon Γ\to \mathbb{R}$ be an additive character. We prove that the ray $[χ]$ does not belong to the BNS set $Σ(Γ)$ if and only if it comes as a pullback along an algebraic fibration $f \colon X \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. We also prove that if $π_1(X)$ admits a solvable quotient which is not virtually nilpotent, there exists a finite étale cover $X_1 \to X$ and a fibration $f \colon X_1 \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. Both of these results were proved by Delzant in the case when $X$ is a compact Kähler manifold. We deduce that $Γ$ is virtually solvable if and only if it is virtually nilpotent, generalising the theorems of Delzant and Arapura-Nori. As a byproduct, we prove a version of Simpson's Lefschetz Theorem for the integral leaves of logarithmic $1$-forms that do not extend to any partial compactification. We give two applications of our results. First, we strengthen the recent theorem of Cadorel-Deng-Yamanoi on virtual nilpotency of fundamental groups of quasi-projective $h$-special and weakly special manifolds. Second, we prove the sharpness of Suciu's tropical bound for the fundamental groups of smooth quasi-projective varieties and answer a question of Suciu on the topology of hyperplane arrangements. |
| title | The Bieri-Neumann-Strebel sets of quasi-projective groups |
| topic | Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2408.06250 |