On Chow rings of quiver moduli

TL;DR

This paper advances the understanding of the Chow ring structure of quiver moduli spaces, enabling computation of integrals and invariants, and establishes geometric relations for specific Kronecker moduli spaces.

Contribution

It describes the point and Todd classes in the Chow ring of quiver moduli, constructs a universal morphism, and relates it to the Kodaira-Spencer morphism, facilitating integral computations.

Findings

Computed invariants of small Kronecker moduli spaces

Established isomorphism of a 6-dimensional Kronecker moduli space with a zero locus on a Grassmannian

Provided a method to compute integrals on 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 $\operatorname{Gr}(2,8)$.