Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus
We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in $\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4$. This includes every ample P -polarized K3 surface, including all Picard jumps, for $P=\mathbb U\oplus D_8(-1)\oplus D_4(-1)$. On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective…
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