Linearization of Quotient Families

J. Math. Sci. Univ. Tokyo
Vol. 26 (2019), No. 3, Page 361-389.

Takamura, Shigeru
Linearization of Quotient Families
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Degeneration of Riemann surfaces is a subject studied by many researchers from various viewpoints - our viewpoint herein is the mixture of algebro-geometric one and topological one. In the present paper, motivated by degenerations of Riemann surfaces, we take the next step towards working in a wider context: after introducing the notion of linear quotient family, we show a linear approximation theorem. Consider a proper submersion between manifolds on which a Lie group (or a discrete group, a finite group) acts equivariantly and properly such that every stabilizer is finite. We show that the quotient of this submersion under the group action is locally orbi-diffeomorphic to a linear quotient family (Linearization Theorem). This has an application to universal families over various moduli spaces (e.g. of Riemann surfaces), enabling us to determine the configuration of singular fibers in universal families and describe how they crash, simply by means of linear algebra and group action.

Keywords: Riemann surface, complex variety, degeneration, monodromy, group action, representation, moduli space, universal family.

Mathematics Subject Classification (2010): Primary 14D05, 14H37, 30F30; Secondary 14H15.
Mathematical Reviews Number: MR4246748

Received: 2018-10-25