Let X be a smooth proper complex $n$-dimensional variety. We consider the cup product map from the product of the Chow groups (modulo torsion) $\mathrm {CH}^{n_1}(\mathrm {V}) \otimes \cdots \otimes \mathrm {CH}^{n_r}(\mathrm {V}) \to \mathrm {CH}^n(\mathrm {X})$ where $\sum _{i=1}^{i=r}n_i = n$ and V is a non empty Zarisky open set in X. If it is surjective (modulo torsion), then the corresponding map from the “edge” Hodge groups $\mathrm {H}^{n_1}(\mathrm {X}, \mathcal {O}_\mathrm {X}) \otimes \cdots \otimes \mathrm {H}^{n_r}(\mathrm {X}, \mathcal {O}_\mathrm {X})\to \mathrm {H}^n(\mathrm {X}, \mathcal {O}_\mathrm {X})$ is surjective. We give variants and discuss some problems.