CODIMENSION-2 CYCLES ON UNIRATIONAL COMPLEX VARIETIES

Let X be a projective complex algebraic manifold. After reviewing the main facts concerning the Zariski sheaves H*(A) on X, associated to U --> H*(U-an, A) for A = Z, Q, Z/n, C we show some consequences of the vanishing of H-0(X, H-3(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(Z)) = 0 we give a description of the global sections of H-3(Z/n) as transcendental n-torsion 2-Hodge cycles, i.e., H-0(X, H-3(Z/n)) congruent to(n) (H-2,H-2(X(an), Z)/NSX). Thus, the non-vanishing of H-0(X, H-3(Z/n)) is equivalent with the, existence of a non-algebraic (2,2)-Hedge integral class : we show examples (of Fano 3-folds and conic bundles) for which all integral Hedge cycles are algebraic. Finally, for unirational varieties and conic bundles over surfaces we show that the cycle map CH2(X) --> H-D(4)(X, X(2)) in Deligne-Beilinson cohomology is injective. We raise several questions and some conjectures.