Degeneracy of Schubert varieties. (English) Zbl 0806.14036

Deodhar, Vinay (ed.), Kazhdan-Lusztig theory and related topics. Proceedings of an AMS special session, held May 19-20, 1989 at the University of Chicago, Lake Shore Campus, Chicago, IL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 139, 181-235 (1992).
The main object of the authors’ interest is the multigraded ring \(R(X)\) of a Schubert variety \(X\) in the flag variety \(G/B\), \(G\) denoting a semisimple simply connected Chevalley group defined over a field \(k\), and \(B\) denoting a Borel subgroup containing a maximal \(k\)-split torus \(T\). \(R(X)\) is the direct sum of \(H^ 0 (X,L)\), \(L\) running over all positive line bundles. The main question is whether \(R(X)\) is Cohen-Macaulay. – The paper gives a positive answer in the case \(G = SL(n)\). The result does not pretend to be new, the general case being treated by A. Ramanathan [Invent. Math. 80, 283-294 (1985; Zbl 0541.14039)], S. Ramanan and A. Ramanathan [Invent. Math. 79, 217-224 (1985; Zbl 0553.14023)].
The paper under review is based on a quite different approach. The main idea is to deform \(R(X)\) into a simpler algebra (by successive flat deformations using the explicit basis of \(R(X))\) and to prove that the deformed algebra is Cohen-Macaulay by methods of M. Hochster and J. A. Eagon [Am. J. Math. 93, 1020-1058 (1971; Zbl 0244.13012)]. The methods are announced to be applicable to other varieties; for example, a forthcoming paper by the second named author treats the case of the variety consisting of pairs of rectangular matrices satisfying \(AB = BA = 0\).
For the entire collection see [Zbl 0784.00017].


14M15 Grassmannians, Schubert varieties, flag manifolds
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13F50 Rings with straightening laws, Hodge algebras
13D10 Deformations and infinitesimal methods in commutative ring theory