论文标题
具有常规单曲的角色品种的算术几何形状
Arithmetic geometry of character varieties with regular monodromy
论文作者
论文摘要
我们研究角色品种作为定位表面组表示的模量，成为还原性的$ g $。我们首先表明，如果$ g/z $在表示形式上自由作用，那么表示形式和角色品种都是光滑且等价的。接下来，我们计算一个流畅的品种家族的要点。也就是说，那些涉及常规半神经和常规的单能单肌。特别是，我们表明这些品种是多项式计数，并获得其$ e $ polynomials的明确表达。最后，通过分析$ e $ polynomial，我们确定了这些品种的某些拓扑不变性，例如欧拉（Euler）特征和连接组件的数量。作为应用程序，我们举例说明了同一个僵化的表示，该表示不是身体上的僵化。
We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.
