# Categorical Topology by H. Herrlich, G. Preuß By H. Herrlich, G. Preuß

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97 (1989), 485-522. W. HESSELINK, Desingularization of varieties of null forms. Invent. Math. 55 (1979), 141-163. Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. Journal 68 (1992), 151-183. Y. Hu, (W, R) matroids and thin Schubert-type cells attached to algebraic torus actions, Proc. of Amer. Math. Soc. 123 No. 9 (1995), 2607-2617. [Ke] [KN] [Kil] G. KEMPF, Instability in invariant theory, Ann. of Math. 108 (1978), 299-316. G. KEMPF and L. NESS, The length of vectors in representation spaces, in « Algebraic geometry, Copenhagen 1978 », Lecture Notes in Math.

A p } without specifying the sub-index (cf. 3). The convention also extends to X and X~\ and so on. 4. Proposition. 1 and the previous assumption. , (^ }, let p^ : Y^ -> Z^, p_ : Y^ -> 7^ be the two natural projections. The subgroup ofG that preserves the fiber of p^ over z e 7^ is G^-U(X) (resp. G,-U(X~ 1 )). Proof. — We shall consider only the map p^.. The other map is treated similarly. First, one checks easily that G^U(X) preserves the fiber p^{ z). 3 (i) any element of G which preserves the fiber must belong to P(X).

To prove (ii), it suffices to consider any particular stratum S^. Let X be the corresponding one-parameter subgroup and p_ : Y™ -> Z^, p^ : Y^ — Z^ be the two natural projections. Note that for any point z e Z^{== Z^), G°, = X(C'). Let us denote the fiber p^^z) by V^. ), G^ acts on V^ (resp. on V~") linearly with positive (resp. negative) weights. We need only to consider V^; the other set can be treated similarly. Now consider any non-stable closed orbit G ' ^ C X ^ F ) where z eZ^ for some a.