The algebraicity conjecture for the Kuga–Satake correspondence

Let $S$ be a polarized K3 surface or a hyper-Kähler variety, and let $V=H^2(S,\mathbb{Q})_{\mathrm{prim}}$ be the primitive polarized Hodge structure of K3 type. Let $\mathrm{KS}(S)$ be the associated Kuga–Satake variety, and consider the embedding of Hodge structures

$$\mu'\colon H^2(S,\mathbb{Q})_{\mathrm{prim}}\hookrightarrow H^1(\mathrm{KS}(S),\mathbb{Q})^{\otimes 2}\subset H^2(\mathrm{KS}(S)^2,\mathbb{Q}).$$

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-theoretic Kuga–Satake correspondence is realized by an algebraic correspondence; its status is not resolved in the supplied text.