Publisher's Synopsis
The authors consider the connections among algebraic cycles, Abelian varieties, and stable rationality of smooth projective varieties in positive characteristic. Recently, Voisin constructed two new obstructions to stable rationality for rationally connected complex projective threefolds by giving necessary and sufficient conditions for the existence of a cohomological decomposition of the diagonal. In this paper, the authors show how to extend these obstructions to rationally chain connected threefolds in positive characteristic via $\ell$-adic cohomological decomposition of the diagonal. This requires extending results in Hodge theory regarding intermediate Jacobians and Abel-Jacobi maps to the setting of algebraic representatives. For instance,they show that the algebraic representative for codimension-two cycle classes on a geometrically stably rational threefold admits a canonical auto-duality, which in characteristic zero agrees with the principal polarization on the intermediate Jacobian coming from Hodge theory. As an application, the authors extend a result of Voisin, and show that in characteristic greater than two, a desingularization of a very general quartic double solid with seven nodes does not admit a universal codimension-two cycle class. In the process, they establish some results on the moduli space of nodal degree-four polarized K3 surfaces in positive characteristic.
