Quotient Singularities and Their Resolutions

Abstract

Let the finite group G act holomorphically on the complex n-dimensional manifold M. Then the orbit space M/G is a complex space. This result, which is due to H. Cartan, is presented in §1-§3. For every point ξ∈ M/G the germ (M/G, ξ) is isomorphic to the germ (ℂⁿ /Γ, 0) for a finite subgroup Γ < GL(n, ℂ), and so to the germ (V, 0) of the affine orbit variety V of Γ, which has been introduced in II§9. Germs of this type are called quotient singularities, They have been characterized and classified by D. Prill. But rather than presenting his general results we restrict attention to the finite subgroups of SL(2, ℂ). Then V is regular at every point ≠0 and definitely singular at 0.