Torsion zero-cycles and the Abel-Jacobi map over the real numbers

Abstract

This is a study of the torsion in the Chow group of zero-cycles on a variety over the real numbers. The first section recalls important results from the literature. The rest of the paper is devoted to the study of the Abel–Jacobi map a: A0XAlbXR restricted to torsion subgroups. Using Roitman’s theorem over the complex numbers and a version of Bloch’s cohomological Abel–Jacobi map over the real numbers, it is shown that this map can be described completely in terms of ´etale cohomology. For some examples (products of curves, abelian varieties, certain fibre bundles) the torsion in the kernel and cokernel of the Abel–Jacobi map a is computed explicitly.