Hodge structures of CM-type

TL;DR

This paper proves that all effective Hodge structures of CM-type are realized in the cohomology of CM abelian varieties over C, and shows how this relates the Hodge conjecture for CM abelian varieties to the general Hodge conjecture.

Contribution

It demonstrates that effective Hodge structures of CM-type appear in CM abelian varieties' cohomology and simplifies the proof linking the Hodge conjecture for CM abelian varieties to the general case.

Findings

Effective Hodge structures of CM-type occur in CM abelian varieties' cohomology

A simplified proof of Hazama's theorem relating Hodge conjectures

Implication that Hodge conjecture for CM abelian varieties extends to the general case

Abstract

We show that any effective Hodge structure of CM-type occurs (without having to take a Tate twist) in the cohomology of some CM abelian variety over C. As a consequence we get a simple proof of the theorem (due to Hazama) that the usual Hodge conjecture for the class of all CM abelian varieties implies the general Hodge conjecture for the same class.