Orbifold k\"ahler groups and the shafarevich conjecture for hirzebruch's covering surfaces with equal weights

TL;DR

This paper explores orbifold Kähler groups in relation to the Shafarevich conjecture, demonstrating that universal covers of certain Hirzebruch surfaces are holomorphically convex, thus providing new insights into the conjecture.

Contribution

It proves the holomorphic convexity of universal covers of Hirzebruch's covering surfaces with equal weights, advancing understanding of the Shafarevich conjecture for these cases.

Findings

Universal cover of Hirzebruch's covering surface with equal weights is holomorphically convex.

Provides partial proof of the Shafarevich conjecture for specific orbifold Kähler groups.

Reduces the conjecture to a problem related to the Serre problem.

Abstract

This article is devoted to examples of (orbifold) K\"ahler groups from the perspective of the so-called Shafarevich conjecture on holomorphic convexity. It aims at pointing out that every quasi-projective complex manifold with an 'interesting' fundamental group gives rise to interesting instances of this long-standing open question. Complements of line arrangements are one of the better known classes of quasi-projective complex surfaces with an interesting fundamental group. We solve the corresponding instance of the Shafarevich conjecture partially giving a proof that the universal covering surface of a Hirzebruch's covering surface with equal weights is holomorphically convex. The final section reduces the Shafarevich conjecture to a question related to the Serre problem.