Geodesic
On algebraic cycles on a~fibre product of families of K3-surfaces
Izvestiya. Mathematics , Tome 77 (2013) no. 1, pp. 143-162.

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

We prove the Hodge conjecture and the standard conjecture of Lefschetz type for fibre squares of smooth projective non-isotrivial families of $\mathrm K3$-surfaces over a smooth projective curve under the assumption that the rank of the lattice of transcendental cycles on a generic geometric fibre of the family is an odd prime. We prove the Hodge conjecture for a fibre product of two non-isotrivial families of $\mathrm K3$-surfaces (possibly with degenerations) under the condition that, for every point of the curve, at least one family has non-singular fibre over this point, and the rank of the lattice of transcendental cycles on a generic geometric fibre of one family is odd and not equal to the corresponding rank for the other.
Keywords: Hodge conjecture, $\mathrm K3$-surface.
Mots-clés : standard conjecture of Lefschetz type 
@article{IM2_2013_77_1_a6,
     author = {O. V. Nikol'skaya},
     title = {On algebraic cycles on a~fibre product of families of {K3-surfaces}},
     journal = {Izvestiya. Mathematics },
     pages = {143--162},
     publisher = {mathdoc},
     volume = {77},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a6/}
}
TY  - JOUR
AU  - O. V. Nikol'skaya
TI  - On algebraic cycles on a~fibre product of families of K3-surfaces
JO  - Izvestiya. Mathematics 
PY  - 2013
SP  - 143
EP  - 162
VL  - 77
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a6/
LA  - en
ID  - IM2_2013_77_1_a6
ER  - 
%0 Journal Article
%A O. V. Nikol'skaya
%T On algebraic cycles on a~fibre product of families of K3-surfaces
%J Izvestiya. Mathematics 
%D 2013
%P 143-162
%V 77
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a6/
%G en
%F IM2_2013_77_1_a6
O. V. Nikol'skaya. On algebraic cycles on a~fibre product of families of K3-surfaces. Izvestiya. Mathematics , Tome 77 (2013) no. 1, pp. 143-162. http://geodesic.mathdoc.fr/item/IM2_2013_77_1_a6/

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

[2] N. Burbaki, Gruppy i algebry Li, gl. 1–3: Algebry Li, svobodnye algebry Li i gruppy Li, Mir, M., 1976 ; гл. 4–6: Группы Кокстера и системы Титса. Группы, порожденные отражениями. Системы корней, 1972 ; гл. 7–8: Подалгебры Картана, регулярные элементы, расщепляемые полупростые алгебры Ли, 1978 ; N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chap. I: Algèbres de Lie., Hermann Cie, Paris, 1971 ; Chap. II: Algèbres de Lie libres. Chap. III: Groupes de Lie, 1972 ; Chap. IV: Groupes de Coxeter et systèmes de Tits. Chap. V: Groupes engendrés par des réflexions. Chap. VI: Systèmes de racines, 1968 ; Chap. VII: Sous-algèbres de Cartan, éléments réguliers. Chap. VIII: Algèbres de Lie semi-simples déployées, 1975 | MR | MR | MR | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[3] G. A. Mustafin, “Families of algebraic varieties and invariant cycles”, Math. USSR-Izv., 27:2 (1986), 251–278 | DOI | MR | Zbl

[4] N. Burbaki, Elementy matematiki. Algebra, gl. VII–IX: Moduli, koltsa, formy, Nauka, M., 1966 ; N. Bourbaki, Éléments de mathématique. Algèbre. Chap. VII: Modules sur les anneaux principaux, Hermann, Paris, 1952 ; Chap. VIII: Modules et anneaux semi-simples, 1958 ; Chap. IX: Formes sesquilinéaires et formes quadratiques, 1959 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[5] S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., 12, Academic Press, New York–London, 1962 | MR | Zbl | Zbl

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

[7] S. G. Tankeev, “The arithmetic and geometry of a generic hypersurface section”, Izv. Math., 66:2 (2002), 393–424 | DOI | DOI | MR | Zbl

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

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

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

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

[12] 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

[13] S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds. II”, Izv. Math., 75:5 (2011), 1047–1062 | DOI | DOI | MR | Zbl

[14] S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds”, Izv. Math., 74:1 (2010), 167–187 | DOI | DOI | MR | Zbl

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

[16] M. Okamoto, “On a certain decomposition of 2-dimensional cycles on a product of two algebraic surfaces”, Proc. Japan Acad. Ser. A Math. Sci., 57:6 (1981), 321–325 | DOI | MR | Zbl

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

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