Geodesic
On the standard conjecture of Lefschetz type for complex projective threefolds.~II
Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 1047-1062.

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

We show that Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the operators $\ast$ and $\Lambda$ of Hodge theory holds for all smooth complex projective threefolds of Kodaira dimension $\varkappa3$.
Keywords: complex projective threefold of non-basic type, Friedlander–Mazur conjecture.
Mots-clés : standard conjecture of Lefschetz type 
@article{IM2_2011_75_5_a7,
     author = {S. G. Tankeev},
     title = {On the standard conjecture of {Lefschetz} type for complex projective {threefolds.~II}},
     journal = {Izvestiya. Mathematics },
     pages = {1047--1062},
     publisher = {mathdoc},
     volume = {75},
     number = {5},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a7/}
}
TY  - JOUR
AU  - S. G. Tankeev
TI  - On the standard conjecture of Lefschetz type for complex projective threefolds.~II
JO  - Izvestiya. Mathematics 
PY  - 2011
SP  - 1047
EP  - 1062
VL  - 75
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a7/
LA  - en
ID  - IM2_2011_75_5_a7
ER  - 
%0 Journal Article
%A S. G. Tankeev
%T On the standard conjecture of Lefschetz type for complex projective threefolds.~II
%J Izvestiya. Mathematics 
%D 2011
%P 1047-1062
%V 75
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a7/
%G en
%F IM2_2011_75_5_a7
S. G. Tankeev. On the standard conjecture of Lefschetz type for complex projective threefolds.~II. Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 1047-1062. http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a7/

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

[2] S. L. Kleiman, “The standard conjectures”, Motives, Part I (Seattle, WA, USA, 1991), Proc. Sympos. Pure Math., 55, Amer. Math. Soc., Providence, RI, 1994, 3–20 | MR | Zbl

[3] 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

[4] E. M. Friedlander, B. Mazur, Filtrations on the homology of algebraic varieties, Mem. Amer. Math. Soc., 110, no. 529, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

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

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

[7] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II”, Ann. of Math. (2), 79 (1964), 109–326 | DOI | MR | Zbl

[8] E. Bierstone, P. D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math., 128:2 (1997), 207–302 | 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 | MR | Zbl

[10] S. G. Tankeev, “On the standard conjecture of Lefschetz type for complex projective threefolds”, Izv. Math., 74:1 (2010), 175–196 | DOI | MR | Zbl

[11] 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

[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] Ph. Griffiths, “Hodge Theory and Geometry”, Bull. London Math. Soc., 36:6 (2004), 721–757 | DOI | MR | Zbl

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

[15] W. Schmid, “Variation of Hodge structure: The singularities of the period mapping”, Invent. Math., 22:3–4 (1973), 211–319 | DOI | MR | Zbl

[16] Ph. Griffiths, Topics in transcendental algebraic geometry, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984 | MR | Zbl

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

[18] 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

[19] 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

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

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

[22] A. Ash, D. Mumford, M. Rapoport, Y. Tai, Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications, v. IV, Math. Sci. Press, Brookline, MA, 1975 | MR | Zbl

[23] K. Kodaira, “On compact analytic surfaces. II”, Ann. of Math. (2), 77:3 (1963), 563–626 | DOI | MR | Zbl

[24] R. Khartskhorn, Algebraicheskaya geometriya, Mir, M., 1981 ; R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York–Heidelberg–Berlin, 1977 | MR | Zbl | MR | Zbl

[25] Y. Miyaoka, “On the Kodaira dimension of minimal threefolds”, Math. Ann., 281:2 (1988), 325–332 | DOI | MR | Zbl

[26] F. Campana, “Orbifolds, special varieties and classification theory”, Ann. Inst. Fourier (Grenoble), 54:3 (2004), 499–630 | MR | Zbl

[27] S. Mori, “Flip theorem and the existence of minimal models for 3-folds”, J. Amer. Math. Soc., 1:1 (1988), 117–253 | DOI | MR | Zbl

[28] M. Reid, “Minimal models of canonical $3$-folds”, Algebraic varieties and analytic varieties (Tokyo, 1981), v. 1, Adv. Stud. Pure Math., North-Holland, Amsterdam, 1981, 131–180 | MR | Zbl

[29] S. Mori, “Classification of higher-dimensional varieties”, Algebraic geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Amer. Math. Soc., Providence, RI, 1987, 269–331 | MR | Zbl

[30] Y. Kawamata, “Abundance theorem for minimal threefolds”, Invent. Math., 108:2 (1992), 229–246 | DOI | MR | Zbl

[31] J. Kollár, “Higher direct images of dualizing sheaves. I”, Ann. of Math. (2), 123:1 (1986), 11–42 | DOI | MR | Zbl

[32] M. Reid, “Young person's guide to canonical singularities”, Algebraic geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Amer. Math. Soc., Providence, RI, 1987, 345–414 | MR | Zbl

[33] R. Elkik, “Rationalité des singularités canoniques”, Invent. Math., 64:1 (1981), 1–6 | DOI | MR | Zbl

[34] Y. Kawamata, “On the plurigenera of minimal algebraic 3-folds with $K\equiv 0$”, Math. Ann., 275:4 (1986), 539–546 | DOI | MR | Zbl

[35] V. V. Shokurov, “$3$-fold log-flips”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 95–202 | MR | Zbl

[36] V. A. Iskovskikh, “Singularities on minimal models of algebraic varieties”, J. Math. Sci. (New York), 106:5 (2001), 3269–3285 | DOI | MR | Zbl

[37] Yu. G. Prokhorov, Induktivnye metody v teorii minimalnykh modelei, Dis. ... dokt. fiz.-matem. nauk, MGU, M., 2001

[38] W. V. D. Hodge, “The topological invariants of algebraic varieties”, Proc. of the International Congress of Mathematicians (Cambridge, MA, 1950), v. 1, Amer. Math. Soc., Providence, RI, 1952, 182–192 | MR | Zbl

[39] A. Grothendieck, “Hodge's general conjecture is false for trivial reasons”, Topology, 8:3 (1969), 299–303 | DOI | MR | Zbl

[40] E. M. Friedlander, B. Mazur, “Correspondence homomorphisms for singular varieties”, Ann. Inst. Fourier (Grenoble), 44:3 (1994), 703–727 | MR | Zbl

[41] E. M. Friedlander, “Filtrations on algebraic cycles and homology”, Ann. Sci. École Norm. Sup. (4), 28:3 (1995), 317–343 | MR | Zbl