Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic

Abstract

We 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, we 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, we 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, we 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 fails one of these two new obstructions, while satisfying all of the classical obstructions. More precisely, it does not admit a universal codimension-two cycle class. In the process, we establish some results on the moduli space of nodal degree-four polarized K3 surfaces in positive characteristic.

A functorial approach to regular homomorphisms

“…However, as it turns out, many of the arguments in the relative setting can be reduced to the case of schemes over a field, and to a base change of field statement. Moreover, even in the case of schemes over fields, we can improve some of our previous results over fields [ACV17] using our new framework. In this spirit, our first main result concerns base-change and descent for algebraic representatives along separable field extensions.…”

“…Murre established the existence of algebraic representatives for codimension 2 cycles for smooth projective varieties over an algebraically closed field in [Mur85]. In [ACV17], we showed that such algebraic representatives admit models over any perfect field. The following theorem provides a uniform statement for the case i = 1, 2, dim X, generalizing the results to the relative setting.…”