Geodesic
On algebraic cycles on Abelian varieties. II
Izvestiya. Mathematics, Tome 14 (1980) no. 2, pp. 383-394 
S. G. Tankeev. On algebraic cycles on Abelian varieties. II. Izvestiya. Mathematics, Tome 14 (1980) no. 2, pp. 383-394. http://geodesic.mathdoc.fr/item/IM2_1980_14_2_a9/
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     title = {On~algebraic cycles on {Abelian} {varieties.~II}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Hodge W. V. D., “The topological invariants of algebraic varieties”, Proc. Inter. Cong., v. 1, Camb. Mass., 1950, 182–192 | MR

[2] Mumford D., “Families of abelian varieties”, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, 347–351 | MR

[3] Mumford D., “A note of Shimura's paper “Discontinuous Groups and Abelian Varieties””, Math. Ann., 181 (1969), 345–351 | DOI | MR | Zbl

[4] Tate J., “Algebraic cycles and poles of zeta functions”, Arithmetical Algebraic Geometry, New York, 1965, 93–110 | MR | Zbl

[5] Burbaki N., Gruppy i algebry Li, gl. 1–3, Mir, M., 1976 ; РіР». 4–6, 1972 ; РіР». 7–8, 1978 | MR | MR | Zbl | MR

[6] Tankeev S. G., “Ob algebraicheskikh tsiklakh na abelevykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 42 (1978), 667–696 | MR | Zbl