Journal of the Mathematical Society of Japan

Degenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ball

Ryota HIRAKAWA and Shigeru TAKAMURA

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Abstract

To each of regular polyhedra and a soccer ball, we associate degenerating families (degenerations) of Riemann surfaces. More specifically: To each orientation-preserving automorphism of a regular polyhedron (and also of a soccer ball), we associate a degenerating family of Riemann surfaces whose topological monodromy is the automorphism. The complete classification of such degenerating families is given. Besides, we determine the Euler numbers of their total spaces. Furthermore, we affirmatively solve the compactification problem raised by Mutsuo Oka — we explicitly construct compact fibrations of Riemann surfaces that compactify the above degenerating families. Their singular fibers and Euler numbers are also determined.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1213-1233.

Dates
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1499846524

Digital Object Identifier
doi:10.2969/jmsj/06931213

Mathematical Reviews number (MathSciNet)
MR3685042

Zentralblatt MATH identifier
06786995

Subjects
Primary: 14D06: Fibrations, degenerations
Secondary: 57M99: None of the above, but in this section 58D19: Group actions and symmetry properties

Keywords
degenerating family of Riemann surfaces fibration monodromy regular polyhedron group action automorphism group cyclic quotient singularity

Citation

HIRAKAWA, Ryota; TAKAMURA, Shigeru. Degenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ball. J. Math. Soc. Japan 69 (2017), no. 3, 1213--1233. doi:10.2969/jmsj/06931213. https://projecteuclid.org/euclid.jmsj/1499846524


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