Quotients by 𝐶* and 𝑆𝐿(2,𝐶) actions

Abstract

Let ρ : C ∗ × X → X \rho :{{\mathbf {C}}^{\ast }} \times X \to X be a meromorphic action of C ∗ {{\mathbf {C}}^{\ast }} on a compact normal analytic space. We completely classify C ∗ {{\mathbf {C}}^{\ast }} -invariant open U ⊆ X U \subseteq X with a compact analytic space U / T U/T as a geometric quotient for a wide variety of actions, including all algebraic actions. As one application, we settle affirmatively a conjecture of D \text {D} . Mumford on compact geometric quotients by SL(2 , C ) {\text {SL(2}},{\mathbf {C}}) of Zariski open sets of ( P C 1 ) n {({\mathbf {P}}_{\mathbf {C}}^1)^n} .