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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.10508 |
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| _version_ | 1866917048754372608 |
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| author | Cogolludo-Agustín, José I. Elduque, Eva |
| author_facet | Cogolludo-Agustín, José I. Elduque, Eva |
| contents | Given a connected dense Zariski open set of a compact Kähler manifold $U$, we address the general problem of the existence of surjective holomorphic maps ${F:U\to C}$ to smooth complex quasi-projective curves from properties of $π_1(U)$. It is known that, if such $F$ exists, then there exists a finitely generated normal subgroup $K\trianglelefteqπ_1(U)$ such that $π_1(U)/K$ is isomorphic to a curve orbifold group $G$ (i.e. the orbifold fundamental group of a smooth complex quasi-projective curve endowed with an orbifold structure). In this paper, we address the converse of that statement in the case where the orbifold Euler characteristic of $G$ is negative, finding a (unique) surjective holomorphic map $F:U\to C$ which realizes the quotient $π_1(U)\twoheadrightarrow π_1(U)/K\cong G$ at the level of (orbifold) fundamental groups. We also prove that our theorem is sharp, meaning that the result does not hold for any curve orbifold group with non-negative orbifold Euler characteristic. Furthermore, we apply our main theorem to address Serre's question of which orbifold fundamental groups of smooth quasi-projective curves can be realized as fundamental groups of complements of curves in $\mathbb{P}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric realizability of epimorphisms to curve orbifold groups Cogolludo-Agustín, José I. Elduque, Eva Algebraic Geometry Given a connected dense Zariski open set of a compact Kähler manifold $U$, we address the general problem of the existence of surjective holomorphic maps ${F:U\to C}$ to smooth complex quasi-projective curves from properties of $π_1(U)$. It is known that, if such $F$ exists, then there exists a finitely generated normal subgroup $K\trianglelefteqπ_1(U)$ such that $π_1(U)/K$ is isomorphic to a curve orbifold group $G$ (i.e. the orbifold fundamental group of a smooth complex quasi-projective curve endowed with an orbifold structure). In this paper, we address the converse of that statement in the case where the orbifold Euler characteristic of $G$ is negative, finding a (unique) surjective holomorphic map $F:U\to C$ which realizes the quotient $π_1(U)\twoheadrightarrow π_1(U)/K\cong G$ at the level of (orbifold) fundamental groups. We also prove that our theorem is sharp, meaning that the result does not hold for any curve orbifold group with non-negative orbifold Euler characteristic. Furthermore, we apply our main theorem to address Serre's question of which orbifold fundamental groups of smooth quasi-projective curves can be realized as fundamental groups of complements of curves in $\mathbb{P}^2$. |
| title | Geometric realizability of epimorphisms to curve orbifold groups |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.10508 |