Bloch's conjecture revisited

Abstract

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 $\omega \in H^2(U,\mathbf{C})$ there is a smaller Zariski open $V\subset U$ such that $$\omega {\mid }_V={\omega }^{\prime }+\zeta$$ where ${\omega }^{\prime }\in F^2{H}^2(V,\mathbf{C})$ and $\zeta$ is integral.

4 pages, LaTeX