Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
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arXiv
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| Format: | Preprint |
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2025
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| author | Cadorel, Benoit Deng, Ya Yamanoi, Katsutoshi |
| author_facet | Cadorel, Benoit Deng, Ya Yamanoi, Katsutoshi |
| contents | This paper is Part III of a series of three. We begin by introducing the notion of $h$-special varieties, which can be seen as varieties "chain-connected by the Zariski closures of entire curves." We prove that if $X$ is either a special complex quasi-projective variety in the sense of Campana or an $h$-special variety, then for any linear representation $\varrho:π_1(X)\to \mathrm{GL}_N(\mathbb{C})$, the image $\varrho(π_1(X))$ is virtually nilpotent. We also provide examples showing that this result is sharp, leading to a revised form of Campana's abelianity conjecture for smooth quasi-projective varieties. In addition, we prove a structure theorem for quasi-projective varieties with big and semisimple representations of the fundamental groups, thereby addressing a conjecture by Kollár in 1995. We also construct several examples of quasi-projective varieties that are special and $h$-special, highlighting certain atypical properties of the non-compact case in contrast with the projective setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20360 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications Cadorel, Benoit Deng, Ya Yamanoi, Katsutoshi Algebraic Geometry Complex Variables This paper is Part III of a series of three. We begin by introducing the notion of $h$-special varieties, which can be seen as varieties "chain-connected by the Zariski closures of entire curves." We prove that if $X$ is either a special complex quasi-projective variety in the sense of Campana or an $h$-special variety, then for any linear representation $\varrho:π_1(X)\to \mathrm{GL}_N(\mathbb{C})$, the image $\varrho(π_1(X))$ is virtually nilpotent. We also provide examples showing that this result is sharp, leading to a revised form of Campana's abelianity conjecture for smooth quasi-projective varieties. In addition, we prove a structure theorem for quasi-projective varieties with big and semisimple representations of the fundamental groups, thereby addressing a conjecture by Kollár in 1995. We also construct several examples of quasi-projective varieties that are special and $h$-special, highlighting certain atypical properties of the non-compact case in contrast with the projective setting. |
| title | Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2512.20360 |