Ample stability implies rigidity for quiver moduli (with Gianni Petrella) is a new paper that gets rid of an annoying technical condition in our earlier paper Rigidity and Schofield's partial tilting conjecture for quiver moduli (and subsequent papers building on the vanising results in this paper).

Recall that, for a dimension vector $\mathbf{d}$ and a stability parameter $\theta$ with $\theta(\mathbf{d})=0$, we have the following three properties:

  1. Ample stability means that the complement of the stable locus in the space of representations of dimension $\mathbf{d}$ has codimension at least $2$. In this case the moduli space has maximal Picard rank: $\operatorname{rk}\operatorname{Pic}=\#Q_0-1$.
  2. The rigidity inequality asks that, for every nontrivial Harder–Narasimhan type $\mathbf{d}^*=(\mathbf{d}^1,\ldots,\mathbf{d}^\ell)$, where $\mu(\mathbf{e})=\theta(\mathbf{e})/|\mathbf{e}|$ denotes the slope, \[ \mu(\mathbf{d}^1)-\mu(\mathbf{d}^\ell) \lt \sum_{1\leq m\lt n\leq\ell} \bigl(\mu(\mathbf{d}^m)-\mu(\mathbf{d}^n)\bigr) \bigl(-\langle\mathbf{d}^m,\mathbf{d}^n\rangle\bigr). \] This is precisely what is needed to apply Teleman quantization to the bundle $\mathcal{U}_i\otimes\mathcal{U}_j^\vee$. We would like to be able to apply this as widely as possible!
  3. Strong ample stability means that for every proper nonzero subdimension vector $\mathbf{e}$ with $\theta(\mathbf{e})>0$ we have $\langle\mathbf{e},\mathbf{d}-\mathbf{e}\rangle\leq -2$, where $\langle-,-\rangle$ is the Euler form. Strong ample stability implies ample stability, and in fact also implies the rigidity inequalities.

Thus, in the $\theta$-coprime setting, the previously known implications were \begin{equation} \text{ample stability} \Longleftarrow \text{rigidity inequality} \Longleftarrow \text{strong ample stability}. \end{equation} Our new paper proves the converse of the first implication: ample stability implies the rigidity inequality. Consequently, the vanishing results for universal bundles, Schofield's partial tilting conjecture, and rigidity of quiver moduli only require ample stability.

So from now on, strong ample stability is a useful explicit criterion to verify ample stability, but it (or the tedious to verify rigidity inequalities) is no longer needed to apply the results I like to apply. This brings me great relief!