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

TL;DR

This paper extends obstructions to stable rationality from complex to positive characteristic, using cohomological decomposition of the diagonal and properties of intermediate Jacobians, with applications to specific threefolds and K3 surfaces.

Contribution

It introduces new obstructions to stable rationality in positive characteristic by extending Hodge-theoretic results and algebraic cycle analysis, including canonical auto-duality of algebraic representatives.

Findings

Desingularized quartic double solids with seven nodes do not admit universal codimension-two cycles in characteristic > 2.

Extended Voisin's results to positive characteristic for rationally chain connected threefolds.

Established properties of moduli space of nodal degree-four polarized K3 surfaces 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.