Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories
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arXiv
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| Natura: | 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 is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: π_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois conjugate variety $X^σ$ with $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, we prove the generalized Green-Griffiths-Lang conjecture. When $\varrho$ is furthermore large, we show that the special subsets of $X^σ$ describing the non-hyperbolicity locus coincide, and that this locus is proper exactly when $X$ is of log general type. Moreover, if the Zariski closure of $ρ(π_1(X))$ is semisimple, we prove that there exists a proper Zariski closed subset $Z \subsetneq X^σ$ such that every subvariety not contained in $Z$ is of log general type and all entire curves in $X^σ$ are contained in $Z$. This result extends the theorems of the third author (2010) and of Campana-Claudon-Eyssidieux (2015) from projective to quasi-projective varieties, and yields stronger conclusions even in the projective case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories Cadorel, Benoit Deng, Ya Yamanoi, Katsutoshi Algebraic Geometry Complex Variables This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: π_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois conjugate variety $X^σ$ with $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, we prove the generalized Green-Griffiths-Lang conjecture. When $\varrho$ is furthermore large, we show that the special subsets of $X^σ$ describing the non-hyperbolicity locus coincide, and that this locus is proper exactly when $X$ is of log general type. Moreover, if the Zariski closure of $ρ(π_1(X))$ is semisimple, we prove that there exists a proper Zariski closed subset $Z \subsetneq X^σ$ such that every subvariety not contained in $Z$ is of log general type and all entire curves in $X^σ$ are contained in $Z$. This result extends the theorems of the third author (2010) and of Campana-Claudon-Eyssidieux (2015) from projective to quasi-projective varieties, and yields stronger conclusions even in the projective case. |
| title | Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2512.15367 |