Let $X$ be a projective complex algebraic manifold. After reviewing the main facts concerning the Zariski sheaves $\mathcal {H}^*(A)$ on $X$, associated to $U \mapsto H^*(U_{an}, A)$ for $A = \bf {Z, Q, Z}/n, \bf {C}$ we show some consequences of the vanishing of $H ^0 (X,\mathcal {H}^3({\bf Z}))$, e.g., for $X$ unirational or a conic bundle over a surface ; indeed, if $X$ is a $3$-fold and $H^0 (X,H^3({\bf Z})) = 0$, we give a description of the global sections of $\mathcal {H}^3 ({\bf Z}/n)$ as transcendental n-torsion $2$-Hodge cycles, i. e., $H^0(X, \mathcal {H}^3 ({\bf Z}/n))\cong _n(H^{2,2}(X_{an}, {\bf Z})/NS^2(X)).$ Thus, the non-vanishing of $H^0(X, \mathcal {H}^3(Z/n)$ is equivalent with the existence of a non-algebraic $(2, 2)$-Hodge integral : we show examples (of Fano $3$-folds and conic bundles) for which all integral Hodge cycles are algebraic. Finally, for unirational varieties and conic bundles over surfaces we show that the cycle map $CH^2(X)\rightarrow H^4_{\mathcal D}(X,{\bf Z}(2))$ in Deligne-Beilinson cohomology is injective. We raise several questions and some conjectures.