The pure cohomology of multiplicative quiver varieties

TL;DR

This paper proves that the pure cohomology of multiplicative quiver varieties, including genus g twisted character varieties of GL_n, is generated by tautological classes, revealing a deep algebraic structure.

Contribution

It establishes that the pure cohomology of stable loci in multiplicative quiver varieties is generated by tautological classes, extending to twisted character varieties of GL_n.

Findings

Pure cohomology is generated by tautological classes.

Applicable to genus g twisted character varieties of GL_n.

Provides a new understanding of the algebraic structure of these 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 $GL_n$ is generated by tautological classes.