Decomposition of the diagonal, intermediate Jacobians, and universal codimension-2 cycles in positive characteristic - Achter, J.D. (b, 2025)
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.
Introduction -- Preliminaries on algebraic representatives -- Preliminaries on Chow decomposition of the diagonal -- Factoring correspondences with given support -- Chow decomposition and self-duality of the algebraic representative -- Specialization and polarization on the algebraic representative -- Preliminaries on cohomological decomposition of the diagonal -- Cohomological decomposition, torison, algebracity, and the Bloch map -- Cohomological correspondences and algebraic representatives -- Cohomological decomposition and universal codimension-2 cycle classes -- The standard assumption -- Algebraic representatives and cohomological actions of correspondences -- Cohomological decomposition and self-duality of the algebraic representative -- Specialization and polarization on the algebraic representative -- Cohomological decomposition of the diagonal, algebraic representatives, and minimal cohomology classes -- Proof of theorem 2 -- Homological decomposition of the diagonal, resolution of singularities, and specialization -- Quartic double solids and the proof of theorem 3 -- Some facts about Abelian varieties
Includes bibliography
Keywords: Algebraic geometry; Decomposition of the diagonal; Geometry