7 problems

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…

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…

Let $(X,h)$ be a $2n$-dimensional polarized hyper-Kähler variety, and let $H^2(X,\mathbb{Q})_{\mathrm{prim}}$ denote the orthogonal complement of the polarization class in…

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…

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 $F$ be a number field, let $E$ be a coefficient field, and let $\rho \colon \mathcal{G}_{F,E}\to H$ be a motivic Galois representation with $\lambda$-adic realizations…

For two number fields $F$ and $E$, let $\mathcal{M}_{F,E}$ be the category of motives for motivated cycles over $F$ with coefficients in $E$, and let $\mathcal{G}_{F,E}$ be its mot…