44 problems

Mixed-Tate motives are motives belonging to the tannakian subcategory generated by Tate motives, and a period is a number arising from the comparison between algebraic de Rham and…

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…

Let $X$ be a projective nonsingular variety, and define its Griffiths group by … The source identifies this group conjecturally with a motivic extension group involving…

Goncharov's polynomiality conjecture. If $p$ is a prime, then $d_w(p)$ and $d_{w,m}(p)$ are polynomials in $p$.

Goncharov's isomorphism conjecture. The map $\nu_{\bullet,\bullet}^{\cal M}(\mu_N)$ is an isomorphism for either $N=1$, or $N=p$ is a prime and $w=m$.

Let $A$ and $B$ be the varieties appearing above, and consider the mixed motive $H^1(A,A\cap B)$ together with its exact sequence … Here $\mathbb{Q}(0)$ denotes the Tate motive of…

Let $\mathbf{DM}_{\mathrm{gm}}(\mathbb{C};\mathbb{Q})$ be the category of geometric motives over $\mathbb{C}$ with rational coefficients, and let $\mathbf{D}(\mathbb{Q})$ be the de…

Let $V(z)$ denote the polylogarithmic variation of mixed Hodge structure over $\mathbb{P}^1\backslash\{0,1,\infty\}$, with its fibers and extension data described by the polylogari…

Purity, splitness and cellularity conjectures. The following properties should hold:

Let $\mathbfcal M_{g,k}$ denote the logarithmic stack of formal curves with framings, and let $\textsf N^{fr}_{\mathrm{LMM}}$, $\textsf N^{fr}_{\mathrm{MHS}}$, and…

Scholl's conjecture. The property that $[E_{\mathfrak p}]$ has good reduction is independent of the prime $\ell$.

Scholl's conjecture. We expect that:

Pour un corps $F$, soit $\mathrm{DM}(F)$ la catégorie triangulée des motifs à coefficients dans $\mathbb{Q}$, et soit $\mathrm{MT}(F)$ sa catégorie des motifs de Tate mixtes. Pour…

Pour un corps $F$, notons $K_j(F)_\mathbb{Q}$ sa $K$-théorie rationnelle et $\mathrm{gr}^n_\gamma K_j(F)_\mathbb{Q}$ le gradué de poids $n$ pour la filtration gamma. Conjecture de…

Over a number field $K$, let $A$ and $B$ be hyperplane arrangements in general position, and let $\boldsymbol{m}(\mathbb{P}^n\smallsetminus A, A\smallsetminus(A\cap B))$ denote the…

Hodge conjecture for log mixed motives. The realization map is an isomorphism

Independence-of-realizations conjecture. Restriction of realizations induces equivalences of categories

Let $(M,W)$ be a one-parameter mirror pair of Calabi–Yau threefolds, and let $\mathcal{Z}_{\mathrm{MS}}\in\textbf{DM}_{\mathrm{gm}}(\mathbb{Q},\mathbb{Q})$ be the limit mixed motiv…

Let $\mathcal{Z}^{\vee}\in\textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})$ be the dual limit mixed motive, and let $\mathbf{Z}^{\bullet}$ be the complex in…

Let $\mathfrak{R}: \textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})\to D^b(\textbf{MHS}_{\mathbb{Q}})$ be the Hodge realisation functor, and let $\mathbf{P}$ be a mixed Hodge-Tate st…

Let $\mathfrak{R}: \textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})\to D^b(\textbf{MHS}_{\mathbb{Q}})$ be the Hodge realisation functor. A mixed Hodge-Tate structure is a mixed Hodge…

Weakened semisimplicity conjecture $({\rm Semi}_{\leq n}')$. For any integer $d$ and morphism $f:M\rightarrow N$ in ${\rm MM}_{\leq n}$ satisfying either

Park's comparison conjecture for 2-motives. The category of the paper's $2$-motives is a full subcategory of the category of Ayoub's $2$-motives.

Jannsen's weight-filtration conjecture. There is an abelian category ${\rm MM}$ of mixed motives over $k$ such that every mixed motive $M$ has a weight filtration for which…

Let $\mathcal{P}^{\mathfrak{m},+}$ be the ring of effective motivic periods, let $U^{dR}$ be the unipotent radical of the motivic de Rham Galois group, and define … Let…