W. Fulton: Degeneracy Loci and Equivariant Cohomology

posted Mar 15, 2012, 5:20 AM by Tony Shaska   [ updated Apr 10, 2012, 12:16 PM ]

The most basic degeneracy locus is a space of m by n matrices of rank at most r. This is defined by many more equations -- the vanishing of all minors of size r+1 -- than its codimension. Computing degrees of such loci in projective spaces, when the entries are general homogeneous polynomials of suitable degrees, was a challenge to 19th century mathematicians. The challenge increases if the matrix is symmetric or skew-symmetric.  In the 20th century, one considered loci where maps between vector bundles drop rank. Slowly we realized that this problem is essentially  equivalent to computing in equivariant cohomology rings of flag bundles, and that there should be loci and formulas for each element in the corresponding Weyl group. We will discuss recent progress in this problem, especially by Ikeda, Mihalcea, and Naruse, in the equivariant setting.