Hodge Classes on Certain Types of Abelian Varieties

Abstract

The Hodge conjecture for A states that such classes are algebraic in the sense that they are Q-linear combinations of cohomology classes of algebraic subvarieties of codimension p on A. This statement is well known for p = 1 (Lefschetz), and it is trivially true for p = 0. Hence, for a given A, it holds for allp if the ring formed by the Hodge classes under cup product is generated by the Hodge classes of codimension 1. We thus have a simple criterion for the Hodge conjecture to be true for A, which we call the (1,1) criterion. Responding to a question posed by Tate ([14, p. 107]), Mumford found that the (1,1) criterion, while true for all A such that dim(A) c 3, is not necessarily satisfied by abelian varieties of dimension 4. Although Mumford's counterexample involved an abelian variety of CM type [51, Weil [15] has stressed that its most important feature is the action of an imaginary quadratic field k on A in such a way that Lie(A) becomes a free k X C-module. (Abelian varieties of precisely this type play an important role in Deligne's proof [2] that Hodge cycles on an arbitrary abelian variety are absolutely Hodge.) There is speculation that the Hodge conjecture is false, at least in general, for such abelian varieties, but no method is available at present for settling this question either way. Indeed, one knows no example of an algebraic variety for which the Hodge conjecture is false, and only scattered examples (Shioda [12]) of