Duke Mathematical Journal

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

László Fehér and Richárd Rimányi

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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.

Article information

Duke Math. J. Volume 114, Number 2 (2002), 193-213.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14M12: Determinantal varieties [See also 13C40] 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 16G20: Representations of quivers and partially ordered sets


Fehér, László; Rimányi, Richárd. Classes of degeneracy loci for quivers: The Thom polynomial point of view. Duke Math. J. 114 (2002), no. 2, 193--213. doi:10.1215/S0012-7094-02-11421-5. http://projecteuclid.org/euclid.dmj/1087575408.

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