Classes of degeneracy loci for quivers: The Thom polynomial point of view

Abstract

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 $s : E\to F$ is a vector bundle homomorphism. The question is, which cohomology class is defined by the set $\Sigma\sb k(s)\subset M$ 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 $A\sb n$-quivers, and it is more general (can be applied to $A\sb n$-, $D\sb n$-, $E\sb 6$-, $E\sb 7$-, and $E\sb 8$-quivers). We provide sample calculations for $A\sb 3$- and $D\sb 4$-quivers.