Geodesic
On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension
Izvestiya. Mathematics , Tome 69 (2005) no. 1, pp. 143-162.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\pi\colon X\to C$ be a potentially simple complex Abelian scheme of prime relative dimension over a smooth projective curve. We prove that numerical equivalence of algebraic cycles on $X$ coincides with homological equivalence. 
@article{IM2_2005_69_1_a6,
     author = {S. G. Tankeev},
     title = {On the numerical equivalence of algebraic cycles on potentially simple {Abelian} schemes of prime relative dimension},
     journal = {Izvestiya. Mathematics },
     pages = {143--162},
     publisher = {mathdoc},
     volume = {69},
     number = {1},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a6/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension
JO  - Izvestiya. Mathematics 
PY  - 2005
SP  - 143
EP  - 162
VL  - 69
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a6/
LA  - en
ID  - IM2_2005_69_1_a6
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension
%J Izvestiya. Mathematics 
%D 2005
%P 143-162
%V 69
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a6/
%G en
%F IM2_2005_69_1_a6
S. G. Tankeev. On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension. Izvestiya. Mathematics , Tome 69 (2005) no. 1, pp. 143-162. http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a6/

[1] Abdulali S., “Algebraic cycles in families of Abelian varieties”, Can. J. Math., 46:6 (1994), 1121–1134 | MR | Zbl

[2] André Y., “Pour une théorie inconditionnelle des motifs”, Publ. Math. IHES, 83 (1996), 5–49 | MR | Zbl

[3] Borel A., Lineinye algebraicheskie gruppy, Mir, M., 1972 | MR | Zbl

[4] Burbaki N., Gruppy i algebry Li, Gl. 1–3, 1976; Гл. 4–6, 1972 ; Гл. 7–8, Мир, М., 1978 | Zbl | MR

[5] Deligne P., “Théorie de Hodge, II”, Publ. Math. IHES, 40 (1971), 5–57 | MR | Zbl

[6] Deligne P., “La conjecture de Weil pour les surfaces K3”, Invent. Math., 15 (1972), 206–226 | DOI | MR | Zbl

[7] Faltings G., Chai Ch.-L., Degeneration of Abelian varieties, Springer-Verlag, Berlin, 1990 | MR | Zbl

[8] Gordon B. B., “A survey of the Hodge conjecture for Abelian varieties”, Lewis J. D., A survey of the Hodge conjecture, CRM Monograph Series, 10, CRM, Montréal, 1999, 297–356

[9] Griffits F., Kharris Dzh., Printsipy algebraicheskoi geometrii, Mir, M., 1982 | MR

[10] Grothendieck A., “Standard conjectures on algebraic cycles”, Algebr. Geom., Oxford Univ. Press, London, 1969, 193–199 | MR

[11] Hodge W. V. D., “The topological invariants of algebraic varieties”, Proc. Int. Congr. Math., V. 1, 1952, 182–192 | MR | Zbl

[12] Jannsen U., “Motives, numerical equivalence, and semi-simplicity”, Invent. Math., 107 (1992), 447–452 | DOI | MR | Zbl

[13] Kleiman S. L., “Algebraic cycles and the Weil conjectures”, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, 359–386 | MR

[14] Lange H., Birkenhake C., Complex Abelian varieties, Springer-Verlag, Berlin, 1992 | MR

[15] Lieberman D., “Numerical and homological equivalence of algebraic cycles on Hodge manifolds”, Amer. J. Math., 90 (1968), 366–374 | DOI | MR | Zbl

[16] Moonen B. J. J., Zarhin Yu. G., “Hodge classes and Tate classes on simple abelian fourfolds”, Duke Math. J., 77:3 (1995), 553–581 | DOI | MR | Zbl

[17] Mamford D., Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, M., 1968

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

[19] Shimura G., “On analytic families of polarized abelian varieties and automorphic functions”, Ann. Math., 78 (1963), 149–192 | DOI | MR | Zbl

[20] Silverberg A., Zarkhin Yu. G., “Gruppy Khodzha abelevykh mnogoobrazii s chisto multiplikativnoi reduktsiei”, Izv. RAN. Ser. matem., 60:2 (1996), 149–158 | MR | Zbl

[21] Smirnov O. N., “Graded associative algebras and Grothendieck standard conjectures”, Invent. Math., 128 (1997), 201–206 | DOI | MR | Zbl

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

[23] Tankeev S. G., “O standartnoi gipoteze dlya kompleksnykh abelevykh skhem nad gladkimi proektivnymi krivymi”, Izv. RAN. Ser. matem., 67:3 (2003), 183–224 | MR | Zbl

[24] Zarhin Yu. G., “Hodge group of K3 surface”, J. für die reine und angew. Math., 341 (1983), 193–220 | MR

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