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Monodromies of splitting families for degenerations of Riemann surfaces. (English) Zbl 1487.14029

A degeneration of Riemann surfaces is a family of complex curves over an open disk in \(\mathbb{C}\) such that the central fiber is singular and the other fibers are all smooth complex curves. When degenerations of Riemann surfaces are classified from a topological viewpoint, the topological monodromies play a very important role. It has been proved that the topological monodromy of a degeneration is always represented by a pseudo-periodic homeomorphism of negative twist. Matsumoto and Montesinos have shown that conversely, given a pseudo-periodic homeomorphism \(f\) of negative twist, a degeneration with singular fiber whose monodromy homeomorphism coincides with \(f\) up to conjugacy can be constructed. The authors of this paper introduce the concept of “topological monodromies of splitting families” for degenerations of Riemann surfaces, and their “monodromy assortments”. They show that the monodromy assortments of barking families associated with tame simple crusts act as a pseudo-periodic homeomorphism of negative twist on each irreducible component of the main fibers. As an application they show an example of two splitting families for one degeneration that have different topological monodromies, although they give the same splitting.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14H15 Families, moduli of curves (analytic)
14D06 Fibrations, degenerations in algebraic geometry
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References:

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