The pure cohomology of multiplicative quiver varieties

Abstract

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