László M. Fehér

Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces

The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.

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 set6k(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 A n-quivers, and it is more general (can be applied to An-, Dn-, E6-, E7-, and E8-quivers). We provide sample calculations for A 3- and D4-quivers.

On second order Thom-Boardman singularities

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularitiesi,j . The formulas are given as linear combi- nations of Schur polynomials, and all coefficients are nonnegative.

Equivariant classes of matrix matroid varieties

To each subset I of {1, . . . , k} associate an integer r(I). Denote by X the collection of those n × k matrices for which the rank of a union of columns corresponding to a subset I is r(I), for all I. We study the equivariant cohomology class represented by the Zariski closure Y = X. This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of Y , namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces.

Degeneracy of 2-forms and 3-forms

We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation ΛC. Received by the editors January 6, 2004. The first and third authors were partially supported by FKFP0055/2001 and the second author was partially supported by NSF grant DMS-0088950 AMS subject classification: 14N10, 57R45.

The incidence class and the hierarchy of orbits

AbstractR. Rimányi defined the incidence class of two singularities η and ζ as [η]|ζ, the restriction of the Thom polynomial of η to ζ. He conjectured that (under mild conditions) [η]|ζ ≠ 0 ⇔ ζ ⊂

Real solutions of a problem in enumerative geometry

\bar \eta

[98] Topological Aspects of Loop Groups

. Generalizing this notion we define the incidence class of two orbits η and ζ of a representation. We give a sufficient condition (positivity) for ζ to have the property that [η]|ζ ≠ 0 ⇔ ζ ⊂