On~algebraic cycles on Abelian varieties.~II

S. G. Tankeev

Izvestiya. Mathematics , Tome 14 (1980) no. 2, pp. 383-394

Résumé

Let $I$ be a simple 4-dimensional Abelian variety of the first or second type in Albert's classification (i.e. all simple factors of the $\mathbf R$-algebra $[\operatorname{End}I]\otimes_\mathbf Z\mathbf R$ are isomorphic to $\mathbf R$ or $M_2(\mathbf R)$). In this case the algebra $\bigoplus H^{2p}(I,\mathbf Q)\cap H^{p,p}$ over $\mathbf Q$ is generated by divisor classes. If $\dim I=5$, $\operatorname{End}(I)\overset\sim\longrightarrow\mathbf Z$ and the Hodge group $\mathrm{Hg}(I)$ has type $A_1$ or $A_1\times A_1$, then $\dim_\mathbf QH^4(I,\mathbf Q)\cap H^{2,2}=2$ and the $\mathbf Q$-space $H^4(I,\mathbf Q)\cap H^{2,2}$ is not generated by classes of intersections of divisors. Bibliography: 6 titles.