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Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)$
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Paula Tretkoff

Print publication date: 2016

Print ISBN-13: 9780691144771

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691144771.001.0001

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Introduction

Introduction

Chapter:
(p.1) Introduction
Source:
Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
Author(s):

Paula Tretkoff

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691144771.003.0001

This chapter explains that the book deals with quotients of the complex 2-ball yielding finite coverings of the projective plane branched along certain line arrangements. It gives a complete list of the known weighted line arrangements that can produce such ball quotients, and then provides a justification for the existence of the quotients. The Miyaoka-Yau inequality for surfaces of general type, and its analogue for surfaces with an orbifold structure, plays a central role. The book also examines the explicit computation of the proportionality deviation of a complex surface for finite covers of the complex projective plane ramified along certain line arrangements. Candidates for ball quotients among these finite covers arise by choosing weights on the line arrangements such that the proportionality deviation vanishes.

Keywords:   complex 2-ball, finite covering, projective plane, line arrangement, ball quotient, Miyaoka-Yau inequality, orbifold structure, proportionality deviation, complex surface

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