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 $GL_n$ is generated by tautological classes.
Categories: math.AG