The Mumford–Tate conjecture for smooth projective varieties

Let $X$ be a smooth projective variety defined over a finitely generated subfield $F\subset\mathbb{C}$. Under the Artin comparison identification

$$\operatorname{GL}(H^*(X^{\mathrm{an}},\mathbb{Q})\otimes\mathbb{Q}_{\ell})\simeq \operatorname{GL}(H^*_{\operatorname{\acute{e}t}}(X_{\overline{F}},\mathbb{Q}_{\ell})),$$

let $\operatorname{MT}(X)\subset\operatorname{GL}(H^*(X^{\mathrm{an}},\mathbb{Q}))$ be the Mumford–Tate group and let $G_{\ell}(X)^{\circ}$ be the identity component of the Zariski closure of the $\ell$-adic Galois representation. Mumford–Tate conjecture.

$$\operatorname{MT}(X)\otimes\mathbb{Q}_{\ell}=G_{\ell}(X)^{\circ}.$$

This conjecture provides a characteristic-zero strategy for the Tate conjecture by relating the Hodge-theoretic Mumford–Tate group to the arithmetic monodromy group; its general status remains open.