The pure cohomology of multiplicative quiver varieties

Abstract

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers $α_i$, and integers $θ_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a "multiplicative quiver variety" $\mathcal{M}_θ^q(α)$, a trigonometric analogue of the Nakajima quiver variety associated to $Q$, $α$, and $θ$. We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus $\mathcal{M}_θ^q(α)^s$ 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.