Higher-weight Jacobians

Abstract

We define and study Jacobians of Hodge structures with weight greater than 1. A complex variety has a Jacobian of weight 2 precisely when it has maximal Picard number; these weight 2 Jacobians naturally arise in the context of the Brauer group and the Tate conjecture. The case of surfaces of maximal Picard number has been previously studied by Beauville; the higher-dimensional case is also related to the work of Totaro on Hodge structures with no middle pieces. Higher-weight Jacobians are complex tori, and it is generally quite difficult to tell if they are algebraic. However, for abelian varieties of maximal Picard number, we are able to explicitly calculate their higher-weight Jacobians using algebraic number theory, and prove that they are not just abelian varieties but in fact also have maximal Picard number. We also compute higher-weight Jacobians for Kummer varieties and singular K3 surfaces. Via class field theory, we study the field of definition of abelian surfaces and singular K3 surfaces using their weight 2 Jacobians.

v6: shifted generalities and Kummer varieties to new section 4.2; improved exposition and introduction. 30 pages including appendix