Geometric realizability of epimorphisms to curve orbifold groups

TL;DR

This paper establishes conditions under which a surjective holomorphic map from a compact Kähler manifold to a curve exists, based on properties of the fundamental group and orbifold Euler characteristic, with applications to algebraic geometry.

Contribution

It proves the existence of holomorphic maps realizing certain orbifold fundamental groups when the orbifold Euler characteristic is negative, and shows this does not hold otherwise.

Findings

Existence of holomorphic maps for negatively orbifold Euler characteristic groups

Non-existence results for non-negative orbifold Euler characteristic groups

Application to fundamental groups of complements of curves in projective planes

Abstract

Given a connected dense Zariski open set of a compact K\"ahler 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 $\pi_1(U)$. It is known that, if such $F$ exists, then there exists a finitely generated normal subgroup $K\trianglelefteq\pi_1(U)$ such that $\pi_1(U)/K$ 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 $\pi_1(U)\twoheadrightarrow \pi_1(U)/K\cong G$ 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 $\mathbb{P}^2$.