Compact quotients by \({\mathbb{C}}^*\)-actions
Let \(\rho\) : \(T\times X\to X\) be a meromorphic locally linearizable action of \(T={\mathbb{C}}^*\) on an irreducible normal complex space and \(X^ T\) the set of fixed points of \(\rho\). The set \(X^ T\) is non-empty and disconnected. Let \(F_ 1,...,F_ r\) be the connected components of \(X^ T\). The maps \(\phi^{\pm}: X\to X^ T\), \(\phi^{\pm}(x):=\lim_{t^{\pm 1}\to 0} \rho (t,x)\) are well-defined and the sets \(X_ i^{\pm}=\{x\in X| \phi^{\pm}(x)\in F_ i\}\) are constructible (even locally closed analytic sets, if X is a Kähler manifold). A component \(F_ i\) is said to be directly less then a component \(F_ j\) if \(C_{ij}=(X_ i^+\setminus F_ i)\cap (X_ j^-\setminus F_ j)\neq \emptyset\). It is called less than \(F_ j\) \((F_ i<F_ j)\), if there is a sequence \(i=i_ 0,...,i_ k=j\) such that \(F_{i_{\ell}}\) is directly less than \(F_{i_{\ell +1}}\) for \(\ell =0,...,k-1\). A cross section of \(\{\) 1,...,r\(\}\) is a partition of \(\{\) 1,...,r\(\}\) into two non-empty disjoint subsets \(A^-\) and \(A^+\) satisfying \(((j\in A^-\quad and\quad F_ i<F_ j)\Rightarrow i\in A^-).\) A semi cross section of \(\{\) 1,...,r\(\}\) is a division of \(\{\) 1,...,r\(\}\) into three disjoint subsets \(A^-, A^ 0, A^+\), at least two of which are non-empty, which satisfy: \((i)\quad (F_ i<F_ j\quad and\quad j\in A^ 0)\Rightarrow i\not\in A^ 0\); (ii) if \(A^+\neq \emptyset\) then \((A^-\cup A^ 0,A^+)\) is a cross section and if \(A^-\neq \emptyset\) then \((A^-,A^ 0\cup A^+)\) is a cross-section. A set \(B\subset X\) is called semisectional set if \(B=X\setminus (\cup_{i\in A^+}X_ i^+\cup \cup_{j\in A^-}X_ j^-)\) for a semi cross section \((A^-,A^ 0,A^+).\) The main theorem states the following: Let \(U\subset X\) be a Zariski open, T-invariant set. Then U/T is a compact complex space and \(U\to U/T\) is a semi-geometric quotient (i.e. \(U\to U/T\) is a categorical quotient, which is locally Stein), if and only if U is a semi-sectional set corresponding to a semi cross section \((A^-,A^ 0,A^+)\).
scientific article; zbMATH DE number 3903041