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Quotients of $ {\bf C}\sp{m}-\{0\}$ by diagonal $ {\bf C}\sp{\ast} $-actions


Author: Kunio Takijima
Journal: Proc. Amer. Math. Soc. 79 (1980), 581-584
MSC: Primary 32M99; Secondary 14B05
DOI: https://doi.org/10.1090/S0002-9939-1980-0572306-4
MathSciNet review: 572306
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Abstract: Let $ {q_1}, \ldots ,{q_m}$ be positive integers with $ ({q_1}, \ldots ,{q_m}) = 1$ and $ \rho :{{\mathbf{C}}^ \ast } \times {{\mathbf{C}}^m} \to {{\mathbf{C}}^m},\rho (t,{z_1}, \ldots ,{z_m}) = ({t^{{q_1}}}{z_1}, \ldots ,{t^{{q_m}}}{z_m})$ the diagonal $ {{\mathbf{C}}^ \ast }$-action on $ {{\mathbf{C}}^m}$. Then the orbit space $ {{\mathbf{C}}^m} - \{ 0\} /{{\mathbf{C}}^ \ast }$ is a normal analytic space. In this paper, we shall show that $ {{\mathbf{C}}^m} - \{ 0\} /{{\mathbf{C}}^ \ast }$ has only rational singularities and, if $ \delta ({q_1}, \ldots ,{q_m}) \leqslant m - 3$ and $ m \geqslant 3,{{\mathbf{C}}^m} - \{ 0\} /{{\mathbf{C}}^ \ast }$ is rigid, where $ \delta ({q_1}, \ldots ,{q_m})$ is the positive integer defined by $ {q_1}, \ldots ,{q_m}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0572306-4
Keywords: Diagonal $ {{\mathbf{C}}^ \ast }$-action, quotient analytic space, rational singularity, rigid singularity
Article copyright: © Copyright 1980 American Mathematical Society

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