The Hodge conjecture for products of K3 surfaces

Claims the rational Hodge conjecture in every degree for every finite product of projective complex K3 surfaces, with no restriction on Picard numbers, periods or endomorphism fields, including powers of one surface. On the way it re-proves algebraicity of the exact Kuga-Satake correspondence. It says nothing about the integral Hodge conjecture or about hyperkahler varieties beyond what follows.

Withdrawn by the authors. OpenAI's withdrawal notice of 6 October 2026: "The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed rational Hodge theorem for products of projective K3 surfaces unproved." The notice adds that the withdrawal concerns the proof and does not assert that the statement is false. The question is open again. Claims the rational Hodge conjecture in every degree for every finite product of projective complex K3 surfaces, with no restriction on Picard numbers, periods or endomorphism fields, including powers of one surface. On the way it re-proves algebraicity of the exact Kuga-Satake correspondence. It says nothing about the integral Hodge conjecture or about hyperkahler varieties beyond what follows.