5 problems

Motivic control conjecture. The variety $X$ should be motivated by $\mathrm{KS}(X)$, and the motives of $X$ and $Y$ should be isomorphic. These assertions express the expectation t…

Algebraicity conjecture for the Kuga–Satake correspondence. The morphism $\mu'$ is induced by an algebraic cycle on $S\times \mathrm{KS}(S)^2$. This asserts that the Hodge-theoreti…

Kuga--Satake Hodge conjecture. The embedding $\iota_V$ is induced by a correspondence in $A^\ast(X\times KS(V)\times KS(V))$. This is a special case of the Hodge conjecture; the pa…

Let $X$ be a projective hyper-Kähler variety, and let $H^2_{\mathrm{tr}}(X)$ denote its transcendental second cohomology. Let $\mathrm{KS}(X)$ be the Kuga-Satake variety associated…

Let $F$ and $E$ be number fields, and let $H'\twoheadrightarrow H$ be a surjection of linear algebraic groups over $E$ with central torus kernel. Suppose that … is a homomorphism.…