Geometric realizability of epimorphisms to curve orbifold groups

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→ C to smooth complex quasi-projective curves from properties of π₁(U). It is known that, if such F exists, then there exists a finitely generated normal subgroup Kπ₁(U) such that π₁(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→ C which realizes the quotient π₁(U) π₁(U)/K 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 P².