8 problems

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 $R^1\pi_*{\mathbb Q}$ fo…

Let $\mathcal{J}_g$ denote the locus of $g$-dimensional Jacobians in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties, and let…

Tannakian obstruction conjecture. The displayed isomorphism holds. This would complete the obstruction theory for $\mathbf{Rep}(G)$ by identifying every class in…

Let $k$ be an algebraically closed subfield of $\mathbf C$, let $\operatorname{AM}$ be the category of André motives over $k$, let $M\in\operatorname{AM}$, and let $r(M)$ be its re…

Let $\mathcal M^{B}_{un}(Ran(\mathbb A^1))$ and $\mathcal M^{dR}_{un}(Ran(\mathbb A^1))$ be the Betti and de Rham categories equipped with their natural tensor structures and fiber…

Let $k$ be an algebraically closed field of characteristic zero, set $K=k(x)$ with derivation $d/dx$, and let $F_X$ be the abstract free group on a set $X$ of cardinality $|k|$. A…

Pareschi–Popa adjoint-module conjecture. Every summand $\mathscr M\hookrightarrow\mathscr{A}\hspace*{-0.15em}d_\Theta$ is either a skyscraper sheaf or has support $A$.

Assume that $p>0$. A pre-Tannakian category is a $\boldsymbol{k}$-linear, rigid, symmetric monoidal abelian category with finite-dimensional Hom spaces and finite length objects. I…