A useful semistability criterion

Abstract

We provide a short proof for a semistability criterion which is crucial to the construction of master spaces which has drawn interest in recent research in Geometric Invariant Theory. Among the main objects of study of Geometric Invariant Theory are representations of reductive algebraic groups. Given a reductive algebraic group G, defined over a field k, a finite dimensional k-vector space W, and a rational representation p: G GL(W), one obtains an action of G on IP(W), the space of hyperplanes in W, and a representation of G on the algebra R := Sym W. By the Hilbert-Nagata theorem, the algebra RG of G-invariant elements in R is a finitely generated graded subalgebra of R. Therefore, IP(W)//G Proj RG is a projective variety, and there is a rational map 7r: IP(W) --+ IP(W)//G. This is the GIT procedure of forming a quotient. A closed point [w] C EP(W) is called semistable, if 7r is defined in [w], i.e., if there is a non-constant homogeneous element in RG not vanishing in [w]. The set IP(W)88 of semistable points is open, and a point [w] c IP(W) is called polystable, if [w] is semistable and the G-orbit of [w] is closed in P(W)88. The map 7r identifies the closed points of IP(W)//G with the set of G-orbits of polystable points. A central task of GIT is now to identify the semistable and polystable points in IP(W). This is usually achieved by the Hilbert-Mumford criterion [1]. However, if p is the direct sum of other representations, this might become too difficult. In this note, we will provide a short and elementary proof of a theorem from [2], dealing with this situation. More precisely, let k and G be as before, and W1,...,W, finite dimensional k-vector spaces. Suppose we are given representations pi: G GL(Wi), i = 1, ..., s, with W = W1 ... ED W, and p = p, ED * * ffl For any t = (Li, ...,tt) with 0 0. In all these cases, one can safely apply the Hilbert-Mumford criterion. The outcome is that the semistable points correspond to semistable L-oriented pairs, using the formulation of semistability in terms of the parameter dependent semistability concept for Bradlow pairs. This content downloaded from 157.55.39.19 on Tue, 14 Jun 2016 05:30:35 UTC All use subject to http://about.jstor.org/terms A USEFUL SEMISTABILITY CRITERION 1925 PROOF OF THE THEOREM We first prove the semistable part. By definition, the point w is semistable if and only if, for some k > 0, there is an invariant section in H0((9(k)) = SkW which does not vanish in w. Now, SkW = Oki+*+k,=k Ski W0 (g ... S S . Ws and the representation of G on SkW respects this decomposition, i.e., (SkW)G (~kl+--+k =k(SkiWl (... (? SksWs)G. Therefore, w is semistable if and only if we find a section in some (Sk~l W,, o . Sklt Wt )G which does not vanish in w. This is precisely the assertion of the theorem. Note that we have used only the very definition of semistability. The polystable part will follow easily by means of an induction from the following ClaiR. Let w' ([w= 1,v WL2], [WL3], ..., [wjt]) be a point in P(WVV, e W142) X P(13 t. ) Then w' is semistable (polystable) w.r.t. the given linearization in ((k, k3, ..., kt), if and only if either ([WI, [wI 3 ],) ..., [wI J]) is semistable (polystable) in P(.tt) w.r.t. the linearization in O (k, k3I ...., kt) for either i = 1 (and w,2 = 0) or i = 2 (and W,, = 0), or there are positive natural numbers n, k1, and k2, such that k[ +-k2 = nk and the point (Q[W,], [W62], [WL3], ..., [wI ]W1) is semistable (polystable) in P(,1,2,,3.tt) w.r.t. the linearization in O (kj, k2, nk3, ..., nkt). The semistable part is proved as before. First, we suppose that w' is polystable. With the Hilbert-Mumford criterion [1], the claim can be rephrased as follows: There exist non-negative rational numbers K, and K2, not both zero, with fK1 + r/2= k such that for every one-parameter subgroup A of G, we have