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 \({\mathcal {M}}_\theta ^q(\alpha )^{{\text {s}}}\) 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.

对于一个颤音Q和非零标量\（q_i \），非负整数\（\ alpha _i \）和标记每个顶点i的整数\（\ theta _i \）的选择，Crawley-Boevey–Shaw关联了一个乘积颤音品种\（{\ mathcal {M}} _ \ theta ^ q（\ alpha）\），这是与Q，\（\ alpha \）和\（\ theta \）相关的Nakajima箭袋品种的三角类比。我们证明了纯同调，在霍奇-理论意义上来说，稳定的基因座的\（{\ mathcal {M}} _ \ THETA ^ Q（\阿尔法）^ {{\文本{S}}} \）生成作为\（{\ mathbb {Q}} \） -由重言式特征类代数。特别是，\（GL_n \）g扭曲字符属的纯同调是通过重言式类生成的。