# Hodge cycles defined over algebraic extensions of $\mathbf{Q}$

Is it true that the Hodge conjecture for all smooth projective varieties over the complex numbers, follows from the Hodge conjecture for smooth projective varieties defined over $\overline{\mathbf{Q}}$?

Has this reduction been proved? If so, where?

• As far as I know, nothing much has changed since Voisin 2007, Comp. Math: "we consider the question, asked by Maillot and Soule, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied." – anon Feb 3 '18 at 6:10
• I would like to point out my own question mathoverflow.net/questions/289745/…. If what you said were proved, this would mean a drastic reduction in the logical complexity of the Hodge conjecture, and in any case a huge breakthrough. Anyone would have heard about it. – Alex Gavrilov Feb 4 '18 at 10:01