Hodge conjecture in arbitrary codimension

Establish the Hodge conjecture for smooth projective complex varieties in general codimension, namely prove that every rational cohomology class of type (k,k) is a rational linear combination of classes of codimension-k algebraic cycles.

Background

The paper recalls the Hodge conjecture as the characteristic-zero analogue underlying several of its arguments. For a smooth projective complex variety Y, the conjecture concerns rational cohomology classes in H{2k}(Y, Q) that are of Hodge type (k,k), asserting that such classes arise as rational linear combinations of codimension-k algebraic cycles. The paper notes that this conjecture remains unresolved in general, while the codimension-one case is settled by the Lefschetz (1,1)-theorem, even integrally.

References

While this conjecture is open in general, the case $k=1$ is a theorem due to Lefschetz , in fact even integrally:

— The Tate Conjecture for Ordinary Calabi-Yau Threefolds  (2610.13939 - Brantner, 9 Oct 2026) in Section 1, Proofs, paragraph preceding the Lefschetz (1,1)-theorem