Hodge structures an abelian varieties of CM-type (Q2717075)

Let \(X\) be a smooth projective variety over the field of complex numbers. Grothendieck's General Hodge conjecture asserts that the linear span of the set of cohomology classes supported on algebraic subvarieties of codimension at least \(r\) is the largest Hodge structure contained in \(F^rH^n (X,\mathbb{C})\cap H^n(X, \mathbb{Q})\), where \(F^rH^n (X,\mathbb{C})\) denotes the \(r\)-th part of the Hodge filtration. The usual Hodge conjecture coincides with the special case \(n=2r\) of the general Hodge conjecture. The main result of the paper is a proof of the general Hodge conjecture for any complex abelian variety of CM-type such that the Hodge ring of every power of the abelian variety is generated by divisors or equivalently whose Hodge group coincides with its Lefschetz group. The approach of the proof is to reduce the general conjecture to the usual Hodge conjecture. Several examples of such CM-type abelian varieties are given.