On projectivity of good quotients
In this article, the authors study good quotients of semisimple actions on smooth varieties. The main result says the following. Given a connected semisimple group \(G\) acting algebraically on a smooth variety \(X\), suppose that there exists a nonempty open subset which is stable by the action of the normalizer of a maximal torus \(T\), and suppose that a good quotient of \(U\) by the action of \(T\) exists and is projective. Then \(X\) is projective, \(X = GU\) and there exists a \(G\)-linearized ample line bundle \(L\) on \(X\) such that \(U\) is the set of semistable points of \(L\) with respect to \(T\). The authors point out the differences with a similar theorem of Mumford from geometric invariant theory, coming from the fact that in Mumford's theorem, one considers geometric quotients instead of good quotients. The main theorem implies that the intersection of all translates of \(U\) by elements of \(G\) has a projective good quotient with respect to the action of \(G\). The authors conjecture that if the hypotheses are weakened to only assume that the good quotient \(U//T\) exists (and is not necessarily projective) then \(\bigcup_{g \in G} gU//G\) exists in the category of algebraic spaces.
scientific article; zbMATH DE number 827383