On the standard conjecture and the existence of a~Chow--Lefschetz decomposition for complex projective varieties

S. G. Tankeev

Izvestiya. Mathematics , Tome 79 (2015) no. 1, pp. 177-207.

Résumé

We prove the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $*$ and $\Lambda$ of Hodge theory for a smooth complex projective variety $X$ if at least one of the following conditions holds: $X$ is a compactification of the Néron minimal model of an Abelian scheme of relative dimension $3$ over an affine curve, and the generic scheme fibre of the Abelian scheme has reductions of multiplicative type at all infinite places; $X$ is an irreducible holomorphic symplectic (hyperkähler) 4-dimensional variety that coincides with the Altman–Kleiman compactification of the relative Jacobian variety of a family $\mathcal C\to\mathbb P^2$ of hyperelliptic curves of genus 2 with weak degenerations, and the canonical projection $X\to\mathbb P^2$ is a Lagrangian fibration. We also show that a Chow–Lefschetz decomposition exists for every smooth projective 3-dimensional variety $X$ which has the structure of a 1-parameter non-isotrivial family of K3-surfaces (with degenerations) or a family of regular surfaces of arbitrary Kodaira dimension $\varkappa$ with strong degenerations.

Détail

Keywords: Néron minimal model, reduction of multiplicative type, K3-surface, hyperkähler variety
Mots-clés : standard conjecture of Lefschetz type, Chow–Lefschetz decomposition, Abel–Jacobi map.

Bibliographie

[1] 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

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

[3] 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

[4] F. Charles, E. Markman, “The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of $K3$ surfaces”, Compos. Math., 149:3 (2013), 481–494 | DOI | MR | Zbl

[5] O. V. Nikol'skaya, “On algebraic cycles on a fibre product of families of K3-surfaces”, Izv. Math., 77:1 (2013), 143–162 | DOI | DOI | MR | Zbl

[6] Vik. S. Kulikov, “Degenerations of $K3$ surfaces and Enriques surfaces”, Math. USSR-Izv., 11:5 (1977), 957–989 | DOI | MR | Zbl

[7] S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties and compactifications of Néron minimal models”, Izv. Math., 78:1 (2014), 169–200 | DOI | DOI | MR | Zbl

[8] Ch. Vial, “Projectors on the intermediate algebraic Jacobians”, New York J. Math., 19 (2013), 793–822 | MR | Zbl

[9] G. Kempf, F. F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I., Lecture Notes in Math., 339, Springer-Verlag, Berlin–New York, 1973, viii+209 pp. | MR | Zbl

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

[11] Vik. S. Kulikov, P. F. Kurchanov, “Complex algebraic varieties: periods of integrals and Hodge structures”, Algebraic geometry. III, Encyclopaedia Math. Sci., 36, Springer, Berlin, 1998, 1–217 | DOI | MR | MR | Zbl

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

[13] 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

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

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

[16] N. Bourbaki, Éléments de mathématique, Algèbre. Ch. X. Algèbre homologique, Masson, Paris, 1980, vii+216 pp. | MR | MR | Zbl | Zbl

[17] 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

[18] S. G. Tankeev, “On the standard conjecture for complex 4-dimensional elliptic varieties”, Izv. Math., 76:5 (2012), 967–990 | DOI | DOI | MR | Zbl

[19] D. R. Morrison, “The Clemens–Schmid exact sequence and applications”, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984, 101–119 | MR | Zbl

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

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

[22] V. V. Shokurov, “Algebraic curves and their Jacobian”, Algebraic geometry. III, Encyclopaedia Math. Sci., 36, Springer, Berlin, 1998, 219–270 | DOI | MR | MR | Zbl | Zbl

[23] P. Deligne, D. Mumford, “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math., 36:1 (1969), 75–109 | DOI | MR | Zbl | Zbl

[24] N. Burbaki, Gruppy i algebry Li, gl. 1–3, Mir, M., 1976, 496 pp. ; гл. 4–6, 1972, 334 с. ; гл. 7, 8, 1978, 342 с. ; N. Bourbaki, Groupes et algèbres de Lie, Chap. 1, Actualités Sci. Indust., 1285, 2nd éd., Hermann, Paris, 1971, 146 pp. ; Chaps. 2, 3, 1349, 1972, 320 pp. ; Chaps. 4–6, 1337, 1968, 288 pp. ; Chaps. 7, 8, 1364, 1975, 271 pp. | MR | MR | Zbl | MR | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[25] A. Beauville, “Holomorphic symplectic geometry: a problem list”, Complex and differential geometry, Springer Proc. Math., 8, Springer, Heidelberg, 2011, 49–63 | DOI | MR | Zbl

[26] A. Beauville, “Variétés kähleriennes dont la première classe de Chern est nulle”, J. Differential Geom., 18:4 (1983), 755–782 | MR | Zbl

[27] Y. André, “On the Shafarevich and Tate conjectures for hyperkähler varieties”, Math. Ann., 305:1 (1996), 205–248 | DOI | MR | Zbl

[28] D. Matsushita, “On fibre space structures of a projective irreducible symplectic manifold”, Topology, 38:1 (1999), 79–83 ; “Addendum”, Topology, 40:2 (2001), 431–432 | DOI | MR | Zbl | DOI | MR

[29] D. Matsushita, “Higher direct images of dualizing sheaves of Lagrangian fibrations”, Amer. J. Math., 127:2 (2005), 243–259 | DOI | MR | Zbl

[30] D. Markushevich, “Lagrangian families of Jacobians of genus 2 curves”, J. Math. Sci., 82:1 (1996), 3268–3284 | DOI | MR | Zbl

[31] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York–Heidelberg, 1977, xvi+496 pp. | MR | MR | Zbl | Zbl

[32] D. Mumford, Abelian varieties, Tata Inst. Fundam. Res. Stud. Math., 5, Oxford Univ. Press, London, 1970, viii+242 pp. | MR | Zbl | Zbl

[33] M. Raynaud, “Spécialisation du foncteur de Picard”, Inst. Hautes Études Sci. Publ. Math., 38 (1970), 27–76 | MR | Zbl

[34] A. Grothendieck, “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes des schémas. II”, Inst. Hautes Études Sci. Publ. Math., 24:1 (1965), 5–223 | DOI | MR | Zbl

[35] A. Grothendieck, “Technique de descente et théorèmes d'existence en géométrie algébrique. VI. Les schémas de Picard: propriétés générales”, Fondements de la géométrie algébrique, Extraits du Séminaire Bourbaki, 1957–1962, Secrétariat mathématique, Paris, 1962, 23 p. (Séminaire Bourbaki, 14e année, 1961/62, exp. no. 236) | MR | Zbl

[36] S. Yu. Arakelov, “Families of algebraic curves with fixed degeneracies”, Math. USSR-Izv., 5:6 (1971), 1277–1302 | DOI | MR | Zbl

[37] F. Hirzebruch, Topological methods in algebraic geometry, Grundlehren Math. Wiss., 131, Springer-Verlag, New York, 1966, x+232 pp. | MR | Zbl | Zbl

[38] J. S. Milne, Étale cohomology, Princeton Math. Ser., 33, Princeton Univ. Press, Princeton, N.J., 1980, xiii+323 pp. | MR | MR | Zbl | Zbl

[39] Shiing-Shen Chern, Complex manifolds, Textos de Matemática, 5, Instituto de Física e Matemática, Universidade do Recife, Recife, Pernambuco, 1959, v+181 pp. | MR | Zbl | Zbl

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

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

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

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

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

[45] M. Reid, “Young person's guide to canonical singularities”, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 345–414 | MR | Zbl

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

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

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

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

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

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