Higher-weight Jacobians

TL;DR

This paper introduces and explores Jacobians of Hodge structures with weights greater than 1, providing theoretical insights and explicit examples, and demonstrating their algebraicity in various special cases.

Contribution

It defines higher-weight Jacobians, studies their properties, and computes explicit examples, showing their algebraicity in specific classes of varieties.

Findings

Higher-weight Jacobians are complex tori.

All studied examples are algebraic.

Explicit fields of definition are computed.

Abstract

We define and study Jacobians of Hodge structures with weight greater than 1. Jacobians of weight 2 naturally come up in the context of the Brauer group and the Tate conjecture. They were previously studied in a special case by Beauville in his work on surfaces of maximal Picard number, and are 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. After discussing some general theory, we compute numerous examples of Jacobians of various weights for special classes of varieties: abelian varieties of maximal Picard number, Kummer varieties, and singular K3 surfaces. It turns out that all of these Jacobians are algebraic. We compute their fields of definition.