The Hodge conjecture for CM abelian varieties

Claims the rational Hodge conjecture in every codimension for every complex CM abelian variety, and hence for finite products and powers of such varieties. Consequences stated in the same paper: the generalized Hodge conjecture for CM abelian varieties, and via Milne's theorems the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. It does not prove the Hodge conjecture for arbitrary abelian varieties (a remark gives only algebraicity of specialized Hodge classes on good-reduction special fibers) and says nothing about integral Hodge classes. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn.

Claims the rational Hodge conjecture in every codimension for every complex CM abelian variety, and hence for finite products and powers of such varieties. Consequences stated in the same paper: the generalized Hodge conjecture for CM abelian varieties, and via Milne's theorems the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. It does not prove the Hodge conjecture for arbitrary abelian varieties (a remark gives only algebraicity of specialized Hodge classes on good-reduction special fibers) and says nothing about integral Hodge classes. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn.