Any abelian variety over algebraic numbers has a de Rham-Betti group containing G_m, so odd-degree cohomology carries no non-zero dRB classes.
On the Shafarevich and Tate conjectures for hyperk¨ ahler varieties
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Weight of the De Rham-Betti Structures of Abelian Varieties
Any abelian variety over algebraic numbers has a de Rham-Betti group containing G_m, so odd-degree cohomology carries no non-zero dRB classes.