Quotients of $\textbf {C}^{m}-\{0\}$ by diagonal $\textbf {C}^{\ast }$-actions

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 ${q_1}, \ldots ,{q_m}$.