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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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Existence of Ball Quotients Covering Line Arrangements

Existence of Ball Quotients Covering Line Arrangements

Chapter:
(p.126) Chapter Six Existence of Ball Quotients Covering Line Arrangements
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/9780691144771.003.0007

This chapter justifies the assumption that ball quotients covering line arrangements exist. It begins with the general case on the existence of finite covers by ball quotients of weighted configurations, focusing on log-canonical divisors and Euler numbers reflecting the weight data on divisors on the blow-up X of P2 at the singular points of a line arrangement. It then uses the Kähler-Einstein property to prove an inequality between Chern forms that, when integrated, gives the appropriate Miyaoka-Yau inequality. It also discusses orbifolds and b-spaces, weighted line arrangements, the problem of the existence of ball quotient finite coverings, log-terminal singularity and log-canonical singularity, and the proof of the main existence theorem for line arrangements. Finally, it considers the isotropy subgroups of the covering group.

Keywords:   ball quotient, line arrangement, log-canonical divisor, Euler number, weight, Miyaoka-Yau inequality, orbifold, b-space, finite covering, covering group

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