19 problems

Let $Z$ be as in paragraph A16, let $X$ be the corresponding variety, and suppose that $x,y$ are $Z$-points of $X$. Let $s_\mathfrak{p}^*$ denote the specialization functor to the…

Let $k$ be the base field, let ${\mathcal {P}}$ be the class of schemes considered in the source, and let $X\in {\mathcal {P}}$. The monodromy operator is … The associated objects…

Let $({\rm {LM}}\flat)$ and $({\rm {LM}}\flat*)$ be the categories with the same objects and with morphisms defined using, respectively, the Chern class constructions from $K_0$ an…

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

Goncharov's scissor-congruence realization conjecture. The map constructed in theorem 11.18.01.5(c) is an isomorphism.

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

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. 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…

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

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 $\ast\in\{1,\vec{1}\}$, let $m\geq 0$ and $r\in\mathbb Z$, and let ${\mathsf{MEM}}_\ast$ be the corresponding category of universal mixed elliptic motives. The real and $\ell$-…

Let $\mathsf{DM}^\otimes$ be the ${\mathbb{Q}}$-linear symmetric monoidal stable presentable $\infty$-category of mixed motives. A symmetric monoidal stable presentable $\infty$-ca…

For every variety $X$ over ${\mathbb{Q}}$, suppose there is a universal cohomology theory $X\mapsto h^*(X)$ with values in an abelian category ${\mathcal M}{\mathcal M}$ of mixed m…

Degeneration conjecture. The spectral sequence $T_{w_{Chow}}({\mathcal{H}}^{et}_{{\mathbb{Q}_l}},M)$ degenerates at $E_2$ for any $M\in \operatorname{Obj}DM_c(S)$.

Let $E$ be the elliptic curve and let ${\mathcal A}_{{\frak M}(E)}$ be the algebra of mixed elliptic motives equipped with its grading and differential structure. K(pi,1) conjectur…

Let $E$ be the elliptic curve and let ${\mathcal A}_{{\frak M}(E)}$ denote the algebra of mixed elliptic motives considered in the paper, graded by $i$. Cohomological connectedness…