Quotients by 𝐶* and 𝑆𝐿(2,𝐶) actions

Białynicki-Birula, Andrzej; Sommese, Andrew John

doi:10.1090/S0002-9947-1983-0709583-X

Quotients by $\textbf {C}^{\ast }$ and $\textrm {SL}(2,\textbf {C})$ actions
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Trans. Amer. Math. Soc. 279 (1983), 773-800 Request permission

Abstract:

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

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