Towards the classification of atoms of degenerations. I: Splitting criteria via configuration of singular fibers. (English) Zbl 1054.14016

Takamura, Shigeru

J. Math. Soc. Japan 56, No. 1, 115-145 (2004).

A degeneration is a proper surjective holomorphic map \(\pi :M\rightarrow \Delta\) where \(M\) is a smooth complex surface and \(\Delta\) is the unit disk, the fibre over the origin is singular, and any other fibre is a smooth complex curve of genus \(g\geq 1\). A deformation of a degeneration is called a splitting deformation if it induces a splitting of the singular fibre. A degeneration without splitting deformations is called atomic. The paper is the first of a series of papers studying splitting deformations of degenerations of complex curves. It proposes methods for the construction of splitting deformations and criteria used in the classification of atomic degenerations. The criteria are of two types – they are expressed either in terms of the configuration of the singular fibre or in terms of its subdivisors. These criteria are “visible” and the paper contains many figures explaining how the singular fibre is deformed. The present paper focuses on splitting criteria of the first type.

Reviewer: Vladimir P. Kostov (Nice)

MSC:

14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
14H15	Families, moduli of curves (analytic)
32S30	Deformations of complex singularities; vanishing cycles
14J15	Moduli, classification: analytic theory; relations with modular forms

Keywords:

degeneration of complex curves; deformation of complex structures; atomic degeneration; monodromy