12 problems

Let $M=[u:d4da0 \rightarrow G^n]$, with $u(1)=(R_1,\dots,R_n)$, be the 1-motive defined by the complex numbers $p_i$, $q_j$, and $t_{ij}$. Choose $p'_1,\dots,p'_{n'}$ and…

Let $X$ be a separated finite type scheme over $\mathbb{C}$, and let $d$ denote its dimension. A 1-motive realization conjecture asserts that the mixed Hodge structures … and … ari…

Let $M$ be a 1-motive defined over $\overline{\mathbb{Q}}$, let $\Pi_M$ be its period matrix, and let $\mathcal{G}_{\mathrm{mot}}(M)$ be its motivic Galois group. Grothendieck's pe…

Deligne's conjecture. The first homology and cohomology groups of $X$ should admit an algebraic description in terms of 1-motives.

Let $k$ be a perfect field, let $X$ be an algebraic variety over $k$, and let $Y$ be a closed subvariety of $X$. Write $W_*$ for a weight filtration on crystalline cohomology, and…

Let $X$ be a proper scheme over $\mathbb C$, let $\pi:X_\bullet\to X$ be a smooth proper hypercovering, and let $H^{2p+i}(X)$ be Deligne's mixed $\mathbb Q$-Hodge structure on…

Let $\Omega$ be a lattice in $\mathbb{C}$. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent complex numbers, and let $q_1,\dots,q_r,p_1,\dots,p_n$ be $k$-linearly independe…

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…

Let $M$ be a 1-motive defined over $\overline{\mathbb{Q}}$. Let $\mathrm{T}_{\mathrm{dR}}$ and $\mathrm{T}_{\mathrm{H}}$ denote its de Rham and Hodge realizations, let…

Let $M$ be a 1-motive defined over a sub-field $K$ of $\mathbb{C}$. Its periods are the coefficients of the comparison isomorphism between its De Rham and Hodge realizations, and l…

Let $S$ be the base scheme, and let $\mathop{\mathbf{DA}}\nolimits(S)$ be the triangulated category of motives over $S$. The homotopy t-structure is the t-structure generated by th…

Étale 1-motive realization conjecture. These functors should satisfy, functorially in $X$, for every prime $\ell\ne p$,