The Tannakian variation conjecture for graded cohomology of stable G-bundles

Let $G$ be the reductive group and let $C$ be a smooth projective curve varying in the universal curve, with moduli map $\pi:{\mathfrak C}_g\to {\mathfrak M}_g$. Write $Gr_W H^i({\rm Bun}^s_{G,C})$ for the $i$th cohomology of the stable-bundle moduli space equipped with its weight filtration, and let $R^1\pi_*{\mathbb Q}$ be the first higher direct image local system.

Tannakian variation conjecture. The variation of Hodge structure associated to $Gr_W H^i({\rm Bun}^s_{G,C})$ lies in the Tannakian subcategory generated by $R^1\pi_*{\mathbb Q}$ for all $i$. In particular, the Torelli group acts trivially on this space.

This conjecture predicts that all graded pieces of the cohomology variations of stable $G$-bundle moduli are generated, in the Tannakian sense, by the first cohomology of the underlying curves. It is motivated by the purity and Tannakian containment established in the preceding low-degree range, while the statement for all $i$ remains open; the source notes that the corresponding unrestricted variations are expected to be genuinely mixed.