28 problems

Mixed Tate motivic Galois group conjecture. The groupscheme $\mathrm{Gal}_\mathrm{mot}$ is conjectured to be isomorphic to the motivic Galois group of the Tannakian category of mix…

Let $\mathbb{F}$ be an algebraically closed subfield of $\mathbb{C}$, let $\mathbf{NMM}(\mathbb{F})$ be Nori's tannakian category of mixed motives over $\mathbb{F}$, and let $M$ be…

Derived-category conjecture for motives. The whole category $\mathrm{DM}$ should be the derived category of an abelian category, and the realization functors should be exact.

Let $F$ be a number field and let $E$ be a finite extension of $\mathbb{Q}$. Write $\mathcal{M}_{F,E}$ for André's category of motives for motivated cycles, and let…

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…

Deligne's conjecture. The weight graded Lie algebra of $U$ is freely generated by

Let $A$ be the cocycle algebra $N$ associated with the Bloch–Kriz construction, let $A\{1\}$ be its 1-minimal model, and let $\operatorname{Conn}^{0f}_{N\{1\}}$ be the category of…

Fullness conjecture. The functor $\iota_{\mathbb{C}}^{\textup{Hdg}}$ is fully faithful, and its essential image is stable under subquotients. This is known for the Tannakian subcat…

Assume that $G$ is of type A. Let $\widetilde{\mathrm{KP}}_0$ be the full subcategory of the Koszul perverse coherent heart generated under extensions by the objects in…

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…

Let $\mathbb{F}$ be an algebraically closed subfield of $\mathbb{C}$, and let $\mathbbm{1}$ and $\mathbb{Q}(1)$ denote the unit motive and the Tate motive in a good tannakian categ…

Isomorphism conjecture. The map is conjectured to be an isomorphism.

Super-Tannakian obstruction conjecture. The displayed cokernel identifications and left exact sequence hold. This is the super analogue of the Tannakian obstruction theory, express…

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

A mixed motive is intended to include motives of singular varieties. Beilinson–Grothendieck conjecture. There should exist a Tannakian category of mixed motives, including motives…

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 $X$ be a projective hyper-Kähler variety, and let its defect group $P(X)$ measure the failure of its motive to be generated by the relevant degree-two or Kuga--Satake motive. D…

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…

Grothendieck's motivic Galois group conjecture. The preceding motivic and Tannakian description should exist for smooth projective varieties.

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…

Let $M\in\mathcal{MM}_{\mathrm{Nori}}$ be a Nori motive, let $G(M)$ be the Tannaka dual of the rigid tensor subcategory generated by $M$, and let…

Serre's tensor-description conjecture. The group $G_E$ is the subgroup of $\operatorname{GL}(H_{\mathrm{sing},\sigma}(E))$ preserving the tensors corresponding to morphisms

Let $K$ be a field and let $F$ be a field of characteristic $0$. Let $\mathcal{M}_{K,F}$ be the category of pure homological motives over $K$ with coefficients in $F$. Jannsen's se…

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 $A$.

Let $\mathcal{M}$ be the conjectural tannakian category of motives inside $\textnormal{\textbf{DM}}_{\textnormal{gm}}(k)$, and let $\mathcal{M}^{\textnormal{ss}}\subset\mathcal{M}$…