The pure cohomology of multiplicative quiver varieties

Abstract

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers ${\alpha }_i$, and integers ${\theta }_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a "multiplicative quiver variety" ${\mathcal{M}}_{\theta }^q(\alpha )$, a trigonometric analogue of the Nakajima quiver variety associated to $Q$, $\alpha$, and $\theta$. We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus ${\mathcal{M}}_{\theta }^q(\alpha {)}^s$ is generated as a $\mathbb{Q}$-algebra by the tautological characteristic classes. In particular, the pure cohomology of genus $g$ twisted character varieties of $G{L}_n$ is generated by tautological classes.