On Chow rings of quiver moduli

Abstract

We describe the point class and Todd class in the Chow ring of a quiver moduli space, building on a result of Ellingsrud-Str{\o}mme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira-Spencer morphism. We illustrate the results by computing some invariants of some "small" Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker quiver moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^\vee(1)$ on $\operatorname{Gr}(2,8)$.