TORIC TOPOLOGY OF THE COMPLEX GRASSMANN MANIFOLDS

The family of the complex Grassmann manifolds G(n,k) with the canonical action of the torus T-n = T-n and the analogue of the moment map mu: G(n,k) ->Delta(n,)(k) for the hypersimplex Delta(n,) (k), is well known. In this paper we study the structure of the orbit space G(n,k)/T-n by developing the methods of toric geometry and toric topology. We use a subdivision of G(n,k) into the strata W-sigma. Relying on this subdivision we determine all regular and singular points of the moment map mu, introduce the notion of the admissible polytopes P-sigma such that mu(W-sigma) = (P) over dot(sigma) and the notion of the spaces of parameters F-sigma, which together describe W-sigma/T-n as the product (P) over dot(sigma )x F-sigma. To find the appropriate topology for the set boolean OR(sigma)(P) over dot(sigma) X F(sigma )we introduce also the notions of the universal space of parameters (F) over tilde and the virtual spaces of parameters (F) over tilde (sigma) subset of (F) over tilde such that there exist the projections (F) over tilde (sigma)->(F) over tilde (sigma) Having this in mind, we propose a method for the description of the orbit space G(n,k)/T-n. The existence of the action of the symmetric group S-n on G(n,k) simplifies the application of this method. In our previous paper we proved that the orbit space G(4,2)/T-4, which is defined by the canonical T-4-action of complexity 1, is homeomorphic to partial derivative Delta(4,2 )* CP1. We prove in this paper that the orbit space G(5,2)/T-5, which is defined by the canonical T-5-action of complexity 2, is homotopy equivalent to the space which is obtained by attaching the disc D-8 to the space Sigma(RP2)-R-4 by the generator of the group pi(7)(Sigma(RP2)-R-4) = Z(4). In particular, (G(5,2)/G(4,2))/T-5 is homotopy equivalent to partial derivative Delta(5,2 )* CP2. The methods and the results of this paper are very important for the construction of the theory of (2l, q)-manifolds we have been recently developing, and which is concerned with manifolds M-2l with an effective action of the torus T-q , q <= l, and an analogue of the moment map mu: M-2l -> P-q, where P-q is a q-dimensional convex polytope.