Smooth points of T-stable varieties in G/B and the Peterson map

• 成果类型: Article
• 署名作者: Carrell, JB; Kuttler, J
• 署名单位: University of British Columbia; University of Basel
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-002-0256-5
• 发表日期: 2003
• 页码: 353-379
• 关键词: rational smoothness; schubert varieties; poincare-duality; SINGULARITIES

摘要

Let G be a connected semi-simple algebraic group defined over an algebraically closed field k, and let T subset of B subset of P be respectively a maximal torus, a Borel subgroup and a parabolic subgroup of G. Inspired by a beautiful result of Dale Peterson describing the singular locus of a Schubert variety in G/B, we characterize the T-fixed points in the singular locus of an arbitrary irreducible T-stable subvariety of G/P (a T-variety for short). Peterson's result (cf. The Deformation Theorem, 1) says that if k = C, then a Schubert variety X subset of G/B is smooth at a T-fixed point x if and only if it is smooth at every T-fixed point y > x (in the Bruhat-Chevalley order on the fixed point set X-T) and all the limits tau(c) (X, x) = lim(z-x) T-z (X) (z is an element of C\C-T) of the Zariski tangent spaces T-z(X) of X coincide as C varies over the set of all T-stable curves in X with C-T = {x, y), where y > x. Using this, Peterson showed that if G is simply laced (and defined over C), then every rationally smooth point of a Schubert variety in G/B is smooth. More generally, the deformation tau(c) (X, x) is defined for any k-variety X with a T-action provided C is what we call good, i.e. C is a curve of the form C = Tz, where z is a smooth point of X \ X-T and x is an element of C-T. Our first main result (Theorem 1.4) says that if x is an element of X is an attractive fixed point, then X is smooth at x if and only if there exist at least two good C containing x such that tau(c) (X, x) = TE(X, x), where TE(X, x) denotes the span of the tangent lines of the T-stable curves in X containing x. In addition, if X is Cohen-Macaulay at x and tau(c)(X, x) = TE(X, x) for even one good C, then X is smooth at x. Our second main result (Theorem 1.6) says that if X is a T-variety in GIP, where G is simply laced, then tau(c)(X, x) subset of TE(X, x) for each good C. This is not true for general G, but when G has no G(2) factors, then (X, x) is always contained in the linear span of the reduced tangent cone to X at x. These results lead to several descriptions of the smooth fixed points of a T-variety in G/P and, in particular, they give simple proofs of Peterson's results valid for any algebraically closed field. We also show (cf. Example 7.1) that there can exist T-stable subvarieties in G/B, where G is simply laced, which have rationally smooth T-fixed points in their singular loci.