Monoidal transformations and~conjectures on algebraic cycles

S. G. Tankeev

Izvestiya. Mathematics , Tome 71 (2007) no. 3, pp. 629-655.

Résumé

We consider the following conjectures: $\operatorname{Hodge}(X)$, $\operatorname{Tate}(X)$ (over a perfect finitely generated field), Grothendieck's standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the Hodge operator $\ast$, conjecture $D(X)$ on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture $C(X)$ on the algebraicity of Künneth components of the diagonal for smooth complex projective varieties. We show that they are compatible with monoidal transformations: if one of them holds for a smooth projective variety $X$ and a smooth closed subvariety $Y\hookrightarrow X$, then it holds for $X'$, where $f\colon X'\to X$ is the blow up of $X$ along $Y$. All of these conjectures are reduced to the case of rational varieties.

Bibliographie

[1] U. Jannsen, “Motives, numerical equivalence, and semi-simplicity”, Invent. Math., 107:1 (1992), 447–452 | DOI | MR | Zbl

[2] A. J. Scholl, “Classical motives” (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, part 1, Amer. Math. Soc., Providence, RI, 1994, 163–187 | MR | Zbl

[3] Yu. I. Manin, “Sootvetstviya, motivy i monoidalnye preobrazovaniya”, Matem. sb., 77(119):4 (1968), 475–507 | MR | Zbl

[4] Yu. I. Manin, “Lektsii o $K$-funktore v algebraicheskoi geometrii”, UMN, 24:5 (1969), 3–86 | MR | Zbl

[5] W. Fulton, S. Lang, Riemann–Roch algebra, Grundlehren Math. Wiss., 277, Springer-Verlag, New York, 1985 | MR | Zbl

[6] U. Fulton, Teoriya peresechenii, Mir, M., 1989 ; W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1984 | MR | MR | Zbl

[7] J. D. Lewis, A survey of the Hodge conjecture, 2nd ed., CRM Monogr. Ser., 10, Providence, RI, 1999 | MR | Zbl

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

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

[10] F. Griffits, Dzh. Kharris, Printsipy algebraicheskoi geometrii, t. 1, 2, Mir, M., 1982 ; P. Griffiths, J. Harris, Principles of algebraic geometry, Pure Appl. Math., Wiley-Intersci., New York, 1978 | MR | Zbl | MR | Zbl

[11] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II”, Ann. of Math. (2), 79:1 (1964), 109–203 ; 79:2, 205–326 | DOI | MR | Zbl | DOI

[12] 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

[13] S. G. Tankeev, “Tsikly na prostykh abelevykh mnogoobraziyakh prostoi razmernosti”, Izv. AN SSSR. Ser. matem., 46:1 (1982), 155–170 ; S. G. Tankeev, “Cycles on simple abelian varieties of prime dimension”, Math. USSR-Izv., 20:1 (1983), 157–171 | MR | Zbl | DOI

[14] A. Grothendieck, “Standard conjectures on algebraic cycles”, Algebraic Geometry, Internat. Colloq., Tata Inst. Fund. Res. (Bombay, 1968), Oxford Univ. Press, London, 1969, 193–199 | MR | Zbl

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

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

[17] Y. André, “Pour une théorie inconditionnelle des motifs”, Inst. Hautes Études Sci. Publ. Math., 83 (1996), 5–49 | DOI | MR | Zbl

[18] I. Lambek, Koltsa i moduli, Mir, M., 1971 ; J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, MA–Toronto, ON–London, 1966 | MR | Zbl | MR | Zbl

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

[20] A. V. Kelarev, “On the Jacobson radical of graded rings”, Comment. Math. Univ. Carolin., 33:1 (1992), 21–24 | MR | Zbl

[21] S. G. Tankeev, “O chislennoi ekvivalentnosti algebraicheskikh tsiklov na potentsialno prostykh abelevykh skhemakh prostoi otnositelnoi razmernosti”, Izv. RAN. Ser. matem., 69:1 (2005), 145–164 | MR | Zbl

[22] O. N. Smirnov, “Graded associative algebras and Grothendieck standard conjectures”, Invent. Math., 128:1 (1997), 201–206 | DOI | MR | Zbl

[23] J. T. Tate, “Algebraic cycles and poles of zeta functions”, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, New York, 1965, 93–110 | MR | Zbl

[24] J. Tate, “Conjectures on algebraic cycles in $l$-adic cohomology”, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, part 1, Amer. Math. Soc., Providence, RI, 1994, 71–83 | MR | Zbl

[25] N. Katz, “Etude cohomologique des pinceaux de Lefschetz”, Groupes de monodromie en géométrie algébrique, Lecture Notes in Math., 340, 1973, 254–327 | DOI | Zbl

[26] D. Mamford, Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, M., 1968 ; D. Mumford, Lectures on curves on a algebraic surface, Ann. of Math. Stud., 59, Princeton Univ. Press, Princeton, NJ, 1966 | Zbl | MR | Zbl

[27] S. Mori, “Classification of higher-dimensional varieties”, Algebraic geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, part 1, Amer. Math. Soc., Providence, RI, 1987, 269–331 | MR | Zbl