Asian Journal of Mathematics

Volume 22 (2018)

Number 2

Special issue in honor of Ngaiming Mok (1 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Orbifold Kähler groups and the Shafarevich conjecture for Hirzebruch’s covering surfaces with equal weights

Pages: 317 – 332

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n2.a9

Author

Philippe Eyssidieux (Institut Fourier, Université de Grenoble-Alpes, Saint Martin d’Hères, France)

Abstract

This article is devoted to examples of (orbifold) Kähler 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.

Keywords

Kähler orbifold, fundamental group, line arrangements

2010 Mathematics Subject Classification

14F35, 14J29, 32J27, 32Q30

Received 28 November 2016

Accepted 4 July 2017

Published 15 June 2018