The pure cohomology of multiplicative quiver varieties

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 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 M-theta(q) (alpha)(s) is generated as a 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.

Automated summary

This paper discusses the multiplicative quiver variety associated with a quiver, and proves that its pure cohomology is generated by tautological characteristic classes. Particularly, the pure cohomology of genus g twisted character varieties of GL(n) is also generated by tautological classes.