Degenerating families with finite monodromy groups
type: Item
instance of: scholarly article
title: Degenerating families with finite monodromy groups
zbMATH Open document ID: 1466.14014
author: Takayuki Okuda
published in: Kodai Mathematical Journal
publication date: 2021-07-12T00:00:00Z
review text: A degenerating family of Riemann surfaces is a proper surjective holomorphic map \(\pi :M\rightarrow S\) from a smooth complex surface \(M\) to a smooth complex curve \(S\) with connected fibers such that it has at most finitely many singular fibers. In the semi-local case, when \(S\) is the unit disk in \(\mathbb{C}\) and there is a unique singular fiber over the origin, \(\pi\) is a degeneration of Riemann surfaces. If both \(M\) and \(S\) are compact, then \(\pi\) is a global degenerating family. A degenerating family of Riemann surfaces over a Riemann surface defines a monodromy representation, which is a homomorphism from the fundamental group of a punctured surface to the mapping class group. The authors show that, given such a homomorphism, if its image is finite, then there exists an (isotrivial) degenerating family of Riemann surfaces whose monodromy representation coincides with it. They discuss the special sections of such a degenerating family.
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14H15
zbMATH DE Number: 7370867
zbMATH Keywords: degeneration of Riemann surfaces
zbMATH Keywords: holomorphic fibration
zbMATH Keywords: monodromy
zbMATH Keywords: mapping class group
zbMATH Keywords: quotient family
zbMATH Keywords: quotient singularity
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: kmj44101
OpenAlex ID: W3164407219
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DOI: 10.2996/KMJ44101
description: scientific article; zbMATH DE number 7370867