We show that the Torelli group of a closed surface of genus $\ge 3$ acts nontrivially on the rational cohomology of the space of $3$-element subsets of that surface. This implies that for a Riemann surface of genus $\ge 3$, the mixed Hodge structure on the space of its positive, reduced divisors of degree $3$ does in general not split over $\mathbb{Q}$.

中文翻译：

---

Torelli 群对曲面构型空间的作用

我们证明了亏格的闭曲面的 Torelli 群$\ge \mathrm{3个}$不平凡地作用于空间的有理上同调$\mathrm{3个}$- 该表面的元素子集。这意味着对于亏格的黎曼曲面$\ge \mathrm{3个}$, 混合 Hodge 结构在其正的、减少的度因数的空间上$\mathrm{3个}$一般不会分裂$\mathbb{Q}$.