Uniformization of Cyclic Quotients of Multiplicative A-singularities

J. Math. Sci. Univ. Tokyo
Vol. 23 (2016), No. 3, Page 675–726.

Sasaki, Kenjiro; Takamura, Shigeru
Uniformization of Cyclic Quotients of Multiplicative A-singularities
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This work is motivated by the canonical model of degenerations of Riemann surfaces. For a quotient space $A_{d-1} / \Gamma$ of a `multiplicative' $A$-singularity $A_{d-1}$ in $\mathbb{C}^{n+1}$ under a certain cyclic group action $\Gamma$ on $A_{d-1}$, we $\it{explicitly}$ construct a small finite abelian subgroup $G$ of $GL(n,\mathbb{C})$ such that $A_{d-1} / \Gamma \cong \mathbb{C}^n / G$. A resolution of $\mathbb{C}^n / G$ gives a decomposition of the monodromy (a $\it{higher}$-$\it{dimensional}$ $\it{fractional}$ $\it{Dehn}$ $\it{twist}$) of a degeneration $A_{d-1} / \Gamma \to \mathbb{C}$ into subtwists along the exceptional set (it seems that T. Ashikaga's work on resolutions is related to this). Moreover: (1) We give a numerical criterion for a certain subgroup of $GL(n,\mathbb{C})$ to be small. (2) For a certain family of subgroups of $GL(n,\,\mathbb{C})$, we show that if one subgroup of this family is small, then all subgroups of this family are small $\it{equi}$-$\it{smallness}$ $\it{theorem}$).

Keywords: Uniformization of singularity, Monodromy, Degeneration of Riemann surfaces, Fractional Dehn twist, Small group, Pseudo-reflection group

Mathematics Subject Classification (2010): Primary 32Q30; Secondary 14D05
Mathematical Reviews Number: MR3526585

Received: 2015-05-28