Finiteness of the number of maximal open subsets with good quotients
The author proves that a normal complex variety \(X\) with action of a reductive group \(G\) admits only finitely many maximal invariant open subsets \(U \subset X\) having a good quotient. Here \(p : U \to Y\) is called a good quotient if \(Y\) is a complex algebraic space and locally, in the étale topology, \(p\) looks like the classical invariant theory quotient. The proof is divided into three results, each of which is of independent interest: First it is shown that any algebraic prevariety contains only finitely many maximal separated open subvarieties. The second step is a finiteness statement for maximal open sets with possibly non-separated good quotients by torus actions. Finally the maximal open subsets with good quotient for the action of \(G\) are related to such defined by the action of a maximal torus \(T \subset G\). As a direct consequence of the main result, the author obtains finiteness of the number of maximal invariant open sets admitting an algebraic variety as good quotient space. Moreover, for projective varieties, it follows that only finitely many open sets occur as sets of semistable points of \(G\)-linearized line bundles. In the last section, concrete examples are discussed and open problems are formulated. For \(X\) being the projective space, an explicit treatment is given in joint work of the author [\textit{A. Białynicki-Birula} and \textit{J. Świȩcicka}, Transform. Groups 1, No. 3, 153-185 (1996; Zbl 0912.14016)].
scientific article; zbMATH DE number 1268554