Cycles of small codimension on a~simple $2p$- or $4p$-dimensional Abelian variety

S. G. Tankeev

Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1221-1262.

Résumé

Let $J$ be a simple $2p$- or $4p$-dimensional Abelian variety over the field of complex numbers, where $p\ne 5$ is a prime number. Assume that one of the following conditions holds: 1) $\operatorname{Cent\,End}^0(J)$ is a totally real field of degree 1, 2 or 4 over $\mathbb Q$; 2) $J$ is a simple $2p$-dimensional Abelian variety of CM-type $(K,\Phi)$ such that $K/\mathbb Q$ is a normal extension; 3) $J$ is a simple $2p$-dimensional Abelian variety such that $\operatorname{End}^0(J)$ is an imaginary quadratic extension of $\mathbb Q$. Then for every positive integer $r$ the $\mathbb Q$-space $H^{2r}(J,\mathbb Q)\cap H^{r,r}$ is spanned by cohomology classes of intersections of divisors.

Bibliographie

[1] Borovoi M. V., “Gruppa Khodzha i algebra endomorfizmov abeleva mnogoobraziya”, Voprosy teorii grupp i gomologicheskoi algebry, Yaroslavskii gos. un-t, Yaroslavl, 1981, 124–126 | MR

[2] Burbaki N., Gruppy i algebry Li, Gl. 1–3, Mir, M., 1976 ; Гл. 4–6, 1977; Гл. 7, 8, 1978 | MR

[3] Kokseter G. S. M., Mozer U. O. Dzh., Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, M., 1980 | MR

[4] Dodson B., “On the Mumford–Tate group of an abelian variety with complex multiplication”, J. Algebra, 111 (1987), 49–73 | DOI | MR | Zbl

[5] Kubota T., “On the field extension by complex multiplication”, Trans. Amer. Math. Soc., 118 (1965), 113–122 | DOI | MR | Zbl

[6] Mai L., “Lower bounds for the ranks of CM types”, J. Number Theory, 32 (1989), 192–202 | DOI | MR | Zbl

[7] Mumford D., “A note on paper: Shimura, Discontinuous groups and abelian varieties”, Math. Ann., 181:4 (1969), 345–351 | DOI | MR | Zbl

[8] Mamford D., Abelevy mnogoobraziya, Mir, M., 1971

[9] Murty V. K., “Computing the Hodge group of an abelian variety”, Séminaire de Théorie des Nombres, Paris, 1988–1989, 141–158 ; Progr. Math., 91, Birkhäuser Boston, Boston, MA, 1990 | MR | Zbl

[10] Pohlmann H. J., “Algebraic cycles on abelian varieties of complex multiplication type”, Ann. Math., 88:1 (1968), 161–180 | DOI | MR | Zbl

[11] Ribet K. A., “Hodge classes on certain types of abelian varieties”, Amer. J. Math., 105 (1983), 523–538 | DOI | MR | Zbl

[12] Serr Zh.-P., Lineinye predstavleniya konechnykh grupp, Mir, M., 1970 | Zbl

[13] Shimura G., Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, Mir, M., 1973 | MR | Zbl

[14] Tankeev S. G., “Ob algebraicheskikh tsiklakh na poverkhnostyakh i abelevykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 398–434 | MR | Zbl

[15] Tankeev S. G., “Tsikly na prostykh abelevykh mnogoobraziyakh prostoi razmernosti”, Izv. AN SSSR. Ser. matem., 46:1 (1982), 155–170 | MR

[16] Tankeev S. G., “Abelevy mnogoobraziya i obschaya gipoteza Khodzha”, Izv. RAN. Ser. matem., 57:4 (1993), 192–205

[17] Tankeev S. G., “Tsikly na abelevykh mnogoobraziyakh i isklyuchitelnye chisla”, Izv. RAN. Ser. matem., 60:2 (1996), 159–194 | MR | Zbl

[18] Tankeev S. G., “O sledakh Frobeniusa”, Izv. RAN. Ser. matem., 62:1 (1998), 165–200 | MR | Zbl

[19] White S. P., “Sporadic cycles on CM abelian varieties”, Compositio Math., 88 (1993), 123–142 | MR | Zbl

[20] Zarkhin Yu. G., “Vesa prostykh algebr Li v kogomologiyakh algebraicheskikh mnogoobrazii”, Izv. RAN. Ser. matem., 48:2 (1984), 264–304 | MR | Zbl

[21] Zarhin Yu. G., “Linear irreducible Lie algebras and Hodge structures”, Proceedings of the USA–USSR symposium on algebraic geometry (Chicago, June 20 – July 14, 1989), Lecture Notes in Math., 1479, eds. S. Bloch, I. Dolgachev, W. Fulton, 1991, 281–297 | MR | Zbl