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abstract: |
    Kempf \[1976\] studied proper, $G$-equivariant maps from equivariant vector bundles over flag manifolds to $G$-representations $V$, which he called [*collapsings*]{}. We give a simple formula for the $G$-equivariant cohomology class on $V$, or [*multidegree*]{}, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is $1$, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, we can use the desingularization to give also a formula for the $G$-equivariant $K$-class of the image, whose leading term is the multidegree.

    Our application is to quiver loci and quiver polynomials. Let $Q$ be a quiver of finite type (A, D, or E, in arbitrary orientation), and assign a vector space to each vertex. Let ${\mathrm{Hom}}$ denote the (linear) space of representations of $Q$ with these vector spaces. This carries an action of $GL$, the product of the general linear groups of the individual vector spaces. A [*quiver locus*]{} $\Omega$ is the closure in ${\mathrm{Hom}}$ of a $GL$-orbit, and its multidegree is the corresponding 