Classes of Degeneracy Loci for Quivers: the Thom Polynomial Point of View

The Chern classes for degeneracy loci of quivers are natural generalizations of the Thom-Porteous-Giambelli formula. Suppose that E, F are vector bundles over a manifold M and that s : E → F is a vector bundle homomorphism. The question is, which cohomology class is defined by the set 6k(s) ⊂ M consisting of points m where the linear map s(m) has corank k? The answer, due to I. Porteous, is a determinant in terms of Chern classes of the bundles E, F. We can generalize the question by giving more bundles over M and bundle maps among them. The situation can be conveniently coded by an oriented graph, called a quiver, assigning vertices for bundles and arrows for maps. We give a new method for calculating Chern class formulae for degeneracy loci of quivers. We show that for representation-finite quivers this is a special case of the problem of calculating Thom polynomials for group actions. This allows us to apply a method for calculating Thom polynomials developed by the authors. The method— reducing the calculations to solving a system of linear equations—is quite different from the method of A. Buch and W. Fulton developed for calculating Chern class formulae for degeneracy loci of An-quivers, and it is more general (can be applied to An-, Dn-, E6-, E7-, and E8-quivers). We provide sample calculations for A3and D4-quivers.