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

TL;DR

This work develops a positive-characteristic analogue of Voisin’s decomposition-of-the-diagonal framework to obstruct stable rationality for threefolds. It builds a parallel theory of algebraic representatives in codimension two, including an auto-duality result and a canonical polarization, and links diagonal decompositions to cohomological and ℓ-adic phenomena, via the Bloch map and Abel–Jacobi-type constructions in families. The paper shows that in characteristic greater than two, very general quartic double solids with seven nodes fail the universal codimension-2 cycle class obstruction, aligning with Voisin’s complex-analytic results but in positive characteristic. It also establishes results on the moduli of nodal degree-four polarized K3 surfaces and develops specialization techniques for polarizations and diagonal decompositions across fibers, thus extending the scope of stable-rationality obstructions to a broad positive-characteristic setting.

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.

Theorems & Definitions (179)

• Theorem 1: Auto-duality of the algebraic representative
• Theorem 2: Cohomological decomposition of the diagonal
• Theorem 3: Quartic double solids
• Remark 1.1: Connection with Galois-equivariant regular homomorphisms
• Remark 1.2
• Lemma 1.3: beauville83fourier
• Lemma 1.4
• proof
• Proposition 1.5
• proof