Bloch's conjecture revisited

Let $X$ be a non-singular projective complex surface. We can show that Bloch's conjecture (i.e., that if $p_g=0$ then the Albanese kernel vanishes) is equivalent to the following statement: If $p_g(X)=0$ then for any given Zariski open $U\subset X$ and $\omega\in H^2(U,{\bf C})$ there is a smaller Zariski open $V\subset U$ such that $$\omega\mid_V =\omega'+\zeta$$ where $\omega'\in F^2H^2(V,{\bf C})$ and $\zeta$ is integral.