Geodesic
On the standard conjecture of Lefschetz type for complex projective threefolds
Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 167-187.

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

Under certain natural assumptions on cohomology of a complex projective fibred threefold with semi-stable degenerations, we prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $\Lambda$ and $*$. In particular, $B(X)$ is true if at least one of the following conditions holds: 1) the generic fibre of some $1$-parameter holomorphic family $\pi\colon X\to C$ is birationally equivalent to either a ruled surface, an Enriques surface, or a K3-surface, 2) all the fibres of $\pi$ are smooth surfaces of Kodaira dimension $\varkappa\le0$.
Mots-clés : standard conjecture of Lefschetz type. 
@article{IM2_2010_74_1_a3,
     author = {S. G. Tankeev},
     title = {On the standard conjecture of {Lefschetz} type for complex projective threefolds},
     journal = {Izvestiya. Mathematics },
     pages = {167--187},
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a3/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the standard conjecture of Lefschetz type for complex projective threefolds
JO  - Izvestiya. Mathematics 
PY  - 2010
SP  - 167
EP  - 187
VL  - 74
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a3/
LA  - en
ID  - IM2_2010_74_1_a3
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the standard conjecture of Lefschetz type for complex projective threefolds
%J Izvestiya. Mathematics 
%D 2010
%P 167-187
%V 74
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a3/
%G en
%F IM2_2010_74_1_a3
S. G. Tankeev. On the standard conjecture of Lefschetz type for complex projective threefolds. Izvestiya. Mathematics , Tome 74 (2010) no. 1, pp. 167-187. http://geodesic.mathdoc.fr/item/IM2_2010_74_1_a3/

[1] Sh.-Sh. Chern, Complex manifolds, Instituto de Física e Matemática, Recife, 1959 | MR | Zbl | Zbl

[2] A. Grothendieck, “Standard conjectures on algebraic cycles”, Algebraic geometry, Internat. Colloq. (Bombay, 1968), Oxford Univ. Press, London, 1969, 193–199 | MR | Zbl

[3] Ph. Griffiths, J. Harris, Principles of algebraic geometry, Wiley, New York, 1978 | MR | MR | Zbl | Zbl

[4] S. L. Kleiman, “Algebraic cycles and the Weil conjectures”, Dix esposés sur la cohomologie des schémas, North-Holland, Amsterdam; Masson, Paris, 1968, 359–386 | MR | Zbl

[5] S. G. Tankeev, “On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension”, Izv. Math., 69:1 (2005), 143–162 | DOI | MR | Zbl

[6] D. I. Lieberman, “Numerical and homological equivalence of algebraic cycles on Hodge manifolds”, Amer. J. Math., 90:2 (1968), 366–374 | DOI | MR | Zbl

[7] N. M. Katz, W. Messing, “Some consequences of the Riemann hypothesis for varieties over finite fields”, Invent. Math., 23:1 (1974), 73–77 | DOI | MR | Zbl

[8] S. G. Tankeev, “Monoidal transformations and conjectures on algebraic cycles”, Izv. Math., 71:3 (2007), 629–655 | DOI | MR | Zbl

[9] I. R. Shafarevich, B. G. Averbukh, Yu. R. Vainberg, A. B. Zhizhchenko, Yu. I. Manin, B. G. Moishezon, G. N. Tyurina, A. N. Tyurin, Algebraicheskie poverkhnosti, Tr. MIAN, 75, Nauka, M., 1965 | MR | Zbl

[10] B. B. Gordon, “A survey of the Hodge conjecture for Abelian varieties”, Appendix to J. D. Lewis, A survey of the Hodge conjecture, CRM Monogr. Ser., 10, Amer. Math. Soc., Providence, RI, 1999, 297–356 | MR | Zbl

[11] P. Deligne, “Théorie de Hodge. II”, Inst. Hautes Études Sci. Publ. Math., 40:1 (1971), 5–57 | DOI | MR | Zbl

[12] Yu. G. Zarkhin, “Weights of simple Lie algebras in the cohomology of algebraic varieties”, Math. USSR-Izv., 24:2 (1985), 245–281 | DOI | MR | Zbl

[13] N. Burbaki, Gruppy i algebry Li, gl. 1–3, Mir, M., 1976 ; гл. 4–6, 1972 ; гл. 7, 8, 1978 ; N. Bourbaki, Groupes et algèbres de Lie, Chap. 1, Actualités Sci. Indust., 1285, Hermann, Paris, 1971 ; Chaps. 2–3, 1349, 1972 ; Chaps. 4–6, 1337, 1968 ; Chaps. 7–8, 1364, 1975 | MR | MR | Zbl | MR | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[14] S. Zucker, “Hodge theory with degenerating coefficients: $L_2$ cohomology in the Poincaré metric”, Ann. of Math. (2), 109:3 (1979), 415–476 | DOI | MR | Zbl

[15] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958 | MR | MR | Zbl | Zbl

[16] J. S. Milne, Étale cohomology, Princeton Math. Ser., 33, Princeton Univ. Press, Princeton, NJ, 1980 | MR | MR | Zbl | Zbl

[17] P. Deligne, “Théorie de Hodge. III”, Inst. Hautes Études Sci. Publ. Math., 44:1 (1974), 5–77 | DOI | MR | Zbl

[18] N. Bourbaki, Éléments de mathématique, Algèbre. Ch. 10: Algèbre homologique, Masson, Paris, 1980 | MR | MR | Zbl | Zbl

[19] G. Shimura, “Reduction of algebraic varieties with respect to a discrete valuation of the basic field”, Amer. J. Math., 77:1 (1955), 134–176 | DOI | MR | Zbl

[20] P. Deligne, “La conjecture de Weil. I”, Inst. Hautes Études Sci. Publ. Math., 43:1 (1974), 273–307 | DOI | MR | MR | Zbl | Zbl

[21] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, Berlin–Heidelberg–New York, 1966 | MR | Zbl | Zbl

[22] A. Borel, A. Haefliger, “La classe d'homologie fondamentale d'un espace analytique”, Bull. Soc. Math. France, 89 (1961), 461–513 | MR | Zbl

[23] E. H. Spanier, Algebraic topology, McGraw-Hill, New York–Toronto–London, 1966 | MR | MR | Zbl | Zbl

[24] G. E. Bredon, Sheaf theory, McGraw-Hill, New York–Toronto–London, 1967 | MR | Zbl

[25] C. H. Clemens, “Degeneration of Kähler manifolds”, Duke Math. J., 44:2 (1977), 215–290 | DOI | MR | Zbl

[26] S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. Math., 67:3 (2003), 597–635 | DOI | MR | Zbl

[27] G. Kempf, F. F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., 339, Springer-Verlag, Berlin–New York, 1973 | DOI | MR | Zbl

[28] Yu. G. Zarhin, “Hodge groups of $K3$ surfaces”, J. Reine Angew. Math., 341 (1983), 193–220 | MR | Zbl

[29] K. Kodaira, “On the structure of compact complex analytic surfaces. II”, Amer. J. Math., 88:3 (1966), 682–721 | DOI | MR | Zbl

[30] D. Mumford, Abelian varieties, Oxford Univ. Press, London, 1970 | MR | Zbl | Zbl