Universität Konstanz A general Hilbert - Mumford Criterion — Revised

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 indicate how to obtain all invariant open subsets with good quotient for a given G-action on a complete Q-factorial toric variety. 1. Statement of the results 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 these comprise the good G-sets, see [7]. The construction of 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. Świȩcicka settled the case of good T -sets defined by generalized moment functions [2], and the case U = X , see [3]. For G = SL2, several results can be found in [4], [5], and [12]. Usually, one imposes maximality conditions on the good T -set U , e.g. projectivity 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 not a T -saturated subset of some properly larger U ′ having the same properties, compare [14]. We shall assume that X is Q-factorial, i.e. X is normal and for every Weil divisor some multiple is Cartier. In Section 4, we prove: Theorem 1.1. Let a connected reductive group G act on a Q-factorial complex variety X. Let T ⊂ G be a maximal torus and U ⊂ X a (T, 2)-maximal open subset. Then the intersection W (U) of all translates g ·U , g ∈ G, is open in X, there is a good quotient W (U) → W (U)/ G, and W (U) is T -saturated in U . 1991 Mathematics Subject Classification. 14L24,14L30.