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: http://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

Full Text (PDF format)

This research was partially supported by ANR grant ANR-16-CE40-0011-01 HODGEFUN.

Received 28 November 2016

Accepted 4 July 2017

Published 15 June 2018