Hodge classes on certain types of Abelian varieties. (English) Zbl 0586.14003

Ribet, Kenneth A.

Am. J. Math. 105, 523-538 (1983).

Let A be an abelian variety over the field of complex numbers \({\mathbb{C}}\) such that dim A\(>1\) and its endomorphism algebra \(E=End A\otimes {\mathbb{Q}}\) is an imaginary quadratic field. We have two embeddings, \(\sigma\) and \(\tau\), say, of E into \({\mathbb{C}}\). E acts on the Lie algebra Lie(A) of A; let \(n_{\sigma}\sigma +n_{\tau}\tau\) be the character of this action. Here \(n_{\sigma}\), \(n_{\tau}\) are positive integers and \(n_{\sigma}+n_{\tau}=\dim A\). Assume that \(n_{\sigma}\) and \(n_{\tau}\) are relatively prime (e.g., dim A is prime). Under this assumption the author computes explicitly the Hodge group of A in terms of E and proves that all Hodge classes on A are linear combinations of products of classes of divisors. In particular, all Hodge classes are algebraic [the case of prime dim A was treated earlier by S. G. Tankeev in Math. USSR, Izv. 20, 157-171 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.1, 155-170 (1982; Zbl 0587.14005)]. Notice that for K3 surfaces the Hodge group was explicitly computed by the reviewer [J. Reine Angew. Math. 341, 193-220 (1983; Zbl 0506.14034)].

MSC:

14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14K20	Analytic theory of abelian varieties; abelian integrals and differentials
14C20	Divisors, linear systems, invertible sheaves

Keywords:

Hodge conjecture; (1,1) criterion; abelian variety; endomorphism algebra; Hodge group; classes of divisors

Citations:

Zbl 0587.14005; Zbl 0506.14034