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

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

### Citations:

Zbl 0587.14005; Zbl 0506.14034
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