- Mumford Criterion

We provide a Hilbert-Mumford Criterion for actions of reductive groups G on Q-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by G from certain maximal open subsets admitting a good quotient by a maximal torus of G. As an application, we show how to obtain all invariant open subsets with good quotient for a given G-action on a complete Q-factorial toric variety. Let a reductive group G act on a normal complex algebraic variety X. It is a central problem in Geometric Invariant Theory to construct all G-invariant open subsets V ⊂ X admitting a good quotient, i.e. an affine G-invariant morphism V → V / /G onto a complex algebraic space such that locally V / /G is the spectrum of the invariant functions. Let us call these V ⊂ X for the moment the good G-sets. In principle, it suffices to know all good T-sets U ⊂ X for some fixed maximal torus T ⊂ G, because the good G-sets are precisely the G-invariant good T-sets, see [3]. The construction of " maximal " good T-sets is less hard, and in order to gain good G-sets one studies the following question: Let U ⊂ X be a good T-set. When is the intersection W (U) of all translates g ·U , g ∈ G, a good G-set? The classical Hilbert-Mumford Criterion answers this question in the affirmative for sets of T-semistable points of G-linearized ample line bundles. Moreover, A. Bia lynicki-Birula and J. ´ Swi¸ecicka settled in [2] the case of good T-sets defined by generalized moment functions, and in [3] the case U = X, as mentioned before. For G = SL 2 , several results can be found in [4], [5], and [12]. As indicated, one imposes maximality conditions on the good T-set U , e.g. pro-jectivity or completeness of U/ /T. The most general concept is T-maximality: U is not T-saturated in some properly larger good T-set U ′ , where T-saturated means saturated with respect to the quotient map. For complete X and T-maximal U ⊂ X which are invariant under the normalizer N (T), A. Bia lynicki-Birula conjectures that W (U) is a good G-set [1, Conj. 12.1]. We shall settle the case of (T, 2)-maximal subsets. These are good T-sets U ⊂ X such that U/ /T is embeddable into a toric variety, and U is …