Schubert Decomposition for Milnor Fibers of the Varieties of Singular Matrices

Abstract

We consider the varieties of singular $m \times m$ complex matrices which may be either general, symmetric or skew-symmetric (with $m$ even). For these varieties we have shown in another paper that they had compact submanifolds, for the homotopy types of the Milnor fibers which are classical symmetric spaces in the sense of Cartan. In this paper we use these models, combined with results due to a number of authors concerning the decomposition of Lie groups, symmetric spaces via the model together with Iwasawa decomposition to give a cell decomposition of the global Milnor fibers. The decomposition is in terms of unique ordered factorizations of matrices in the Milnor fibers as products of In the case of symmetric or skew-symmetric matrices, this factorization has the form of iterated Cartan conjugacies by pseudo-rotations. The Schubert (the closures of the cells), are images of products of suspensions of projective spaces (complex, real, or quaternionic). For general or skew-symmetric matrices the cycles have fundamental classes, and for symmetric matrices $\mod 2$ classes, giving a basis for the homology, corresponding to the cohomology generators for the symmetric spaces. For general matrices the duals of the cycles are given as explicit monomials in the generators of the exterior cohomology algebra; and for symmetric matrices they are related to Stiefel-Whitney classes of an associated vector bundle. Furthermore, for any matrix singularity of these types the pull-backs of these cohomolgy classes generate a characteristic subalgebra of the cohomology of the Milnor fiber. These results extend to exceptional orbit hypersurfaces, complements and links.