Hodge classes on abelian varieties

TL;DR

This paper demonstrates that Hodge classes on CM-type abelian varieties can be expressed using divisor and split Weil classes, and links the standard conjecture to the Hodge conjecture for these varieties.

Contribution

It establishes a connection between Hodge classes and specific algebraic classes on CM-type abelian varieties, extending previous work by Deligne and André.

Findings

Hodge classes can be expressed in terms of divisor and split Weil classes

The standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties

Provides new insights into the structure of Hodge classes on CM-type abelian varieties

Abstract

We prove, following Deligne and Andr\'e, that the Hodge classes on abelian varieties of CM-type can be expressed in terms of divisor classes and split Weil classes, and we describe some consequences. In particular, we show that Grothendieck's standard conjecture of Lefschetz type implies the Hodge conjecture for abelian varieties (Abdulali, Andr\'e, ...).