Anders Skovsted Buch, Aarhus University
Formulas for quiver varieties

A quiver variety is a general type of degeneracy locus associated to a quiver of vector bundles and bundle maps over a variety. Examples include Schubert varieties in flag varieties and determinantal varieties, and the study of formulas for quiver varieties tends to produce very interesting combinatorics. In work with Fulton, I have proved a formula for the cohomology class of a quiver variety when the underlying quiver is equioriented of type A, and I have later generalized this to a formula for the Grothendieck class of the quiver variety, i.e. the class of its structure sheaf in the Grothendieck ring of the embedding variety. These formulas are stated in terms of integers called quiver coefficients. Knutson, Miller, and Shimozono have shown that the coefficients in the cohomology formula are non-negative. I will speak about a proof that the more general quiver coefficients in the K-theory formula have signs that alternate with codimension. I will also explain a cohomology formula for non-equioriented quiver varieties, which I have recently proved with R. Rimanyi.