I'm really happy to have On Chow rings of quiver moduli on the arXiv now. It is joint with my good friend Hans Franzen, with whom I also wrote (together with others) the topic of the previous "new paper" post.

The motivation is somewhat similar as to the previous post: there are lots of interesting parallels between

  • moduli of semistable quiver representations (or quiver moduli, in short)
  • moduli of semistable vector bundles on curves

As the title suggests, we now care about Chow rings (and cohomology rings) of these moduli spaces. First of all, we'd like an explicit presentation. For moduli of vector bundles on curves, a nice introduction can be found in Alastair King, Peter Newstead: On the cohomology ring of the moduli space of rank 2 vector bundles on a curve.

For quiver moduli, this was done in Hans Franzen: Chow rings of fine quiver moduli are tautologically presented, building upon Alastair King and Charles Walter: On Chow rings of fine moduli spaces of modules.

Todd classes and point classes

But to do calculations, as for instance implemented in the wonderful Schubert2 for partial flag varieties (in type A), one also needs

  • an expression for the point class (so that you know how you are integrating things)
  • an expression for the Todd class, to make Hirzebruch–Riemann–Roch calculations possible

in terms of the given presentation. I don't know where this would be done for moduli of vector bundles on curves. But for quiver moduli it is now done in our new paper.

The trick is to upgrade the usual 4-term sequence of Hom and Ext of quiver representations to a 4-term sequence of vector bundles on the base of a family. This is not new, but I am really happy with how we approached this problem, as I think it highlights how quiver moduli are parallel to more conventional moduli spaces of sheaves.

Another important ingredient, again looking for parallels with more conventional moduli spaces of sheaves, is the Kodaira–Spencer morphism. Its construction was sketched by Dyer and Polishchuk in NC-smooth algebroid thickenings for families of vector bundles and quiver representations, but we give a novel take on it, explaining the role of Atiyah classes.

A conjecture

Our initial motivation was to do some explicit calculations in a specific case. Namely, Kronecker moduli, or Kronecker quiver moduli, are quiver moduli associated to the easiest possible quivers: Kronecker quivers, with 2 vertices and $n$ arrows. To make things interesting we assume $n\geq 3$.

In many cases these Kronecker quiver moduli have an interpretation as a Grassmannian. But not always. The smallest case is the 3-Kronecker quiver, with dimension vector $(2,3)$. It is a 6-dimensional Fano variety of Picard rank 1 and index 3, with Hodge numbers $\mathrm{h}^{p,p}$ given by \[ 1, 1, 3, 3, 3, 1, 1. \]

There is a great interest in describing Fano varieties in terms of zero loci of equivariant vector bundles on partial flag varieties. When prompted with this challenge, Enrico Fatighenti and Fabio Tanturri quickly produced a zero locus with the same properties, using $\mathcal{Q}^\vee(1)$ on $\operatorname{Gr}(2,8)$.

To find further evidence, or disprove that they were the same variety, we wanted to compute the degree of $\mathcal{O}_X(1)$ and the Hilbert series of $\mathcal{O}_X(1)$, for which we needed the point class and Todd class.

Turns out that those numbers agree, thus giving further evidence to the conjecture that they are indeed the same variety! Next up (please take up this challenge!) would then be:

  • trying to (dis)prove the conjecture,
  • find similar descriptions for Kronecker quiver moduli

Et voilà, now you know why I am so happy with this paper, and you know what is in it, without having to read the less colloquially written introduction (or even worse, the paper itself). Of course, you are still cordially invited to do so (and let us know if you have any comments) if you are interested in quiver moduli!