10 problems

Deligne's absoluteness conjecture. For each $\sigma$, every Hodge cycle in $H^{2i}(\bar{X}\times_{\sigma}\operatorname{Spec}\mathbb{C},\mathbb{Q}(i))$ is the component…

Let $X$ be a smooth projective variety over a number field $F$. Let $(s_{\mathbb A_f},s_{\mathrm{dR}})$ be an absolute Hodge class in … Let…

Absolute Hodge conjecture. If this pair comes from a Hodge class in $(H^w_B(X(\mathbb C),\mathbb Q))^{\otimes}$ for one embedding $\sigma:k\hookrightarrow\mathbb C$, then for every…

Let $R_K^{\rm p}$ be the category of polarized realizations, and let $\mathcal V=(V_{\rm DR},(V_l)_l,(V_\sigma)_\sigma,I_{\infty,\sigma},I_{l,\bar\sigma})$ be a pure polarized real…

Let $S$ be the base of the family considered in the paper. An $\ell$-Galois special subvariety is a closed irreducible complex subvariety maximal among those having a fixed generic…

Let $S$ be the base of the family considered in the paper. A special subvariety is a closed irreducible complex subvariety maximal among those having a fixed generic Mumford–Tate g…

Let $S$ be the base of the family considered in the paper, and for a subvariety $Z\subset S$ let $G_Z$ and $G_Z^{AH}$ be its generic Mumford–Tate and generic absolute Mumford–Tate…

Let $X$ be a smooth projective variety over $K$, let $G_X^{AH}$ denote its absolute Mumford–Tate group, and let $G_{\ell,X}$ denote its $\ell$-adic algebraic Galois group. Absolute…

Let $Z\subset S$ be a closed irreducible subvariety of a variation of Hodge structure. Write $G_Z$ for its generic Mumford–Tate group and $G_Z^{AH}$ for its generic dR-absolute Mum…

Let $(\mathbb{V}^{\sigma})_{\sigma}$ be an absolute variation of Hodge structure on a smooth irreducible quasi-projective complex algebraic variety $S$. For a closed irreducible su…