Geodesic
Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$
Sbornik. Mathematics, Tome 211 (2020) no. 2, pp. 161-200 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$, respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $\operatorname{CH}^p(Y)$. We prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general hyperplane section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_\mathrm{van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$. Bibliography: 33 titles.
Keywords: Chow schemes, cubic fourfold hypersurfaces.
Mots-clés : algebraic cycles, $l$-adic étale monodromy, Picard-Lefschetz formulae 
@article{SM_2020_211_2_a0,
     author = {K. Banerjee and V. Guletskiǐ},
     title = {\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in~$\mathbb P^5$},
     journal = {Sbornik. Mathematics},
     pages = {161--200},
     year = {2020},
     volume = {211},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_2_a0/}
}
TY  - JOUR
AU  - K. Banerjee
AU  - V. Guletskiǐ
TI  - Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 161
EP  - 200
VL  - 211
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_2_a0/
LA  - en
ID  - SM_2020_211_2_a0
ER  - 
%0 Journal Article
%A K. Banerjee
%A V. Guletskiǐ
%T Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$
%J Sbornik. Mathematics
%D 2020
%P 161-200
%V 211
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2020_211_2_a0/
%G en
%F SM_2020_211_2_a0
K. Banerjee; V. Guletskiǐ. Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$. Sbornik. Mathematics, Tome 211 (2020) no. 2, pp. 161-200. http://geodesic.mathdoc.fr/item/SM_2020_211_2_a0/

[1] A. Beauville, “Variétés de Prym et jacobiennes intermédiaires”, Ann. Sci. École Norm. Sup. (4), 10:3 (1977), 309–391 | DOI | MR | Zbl

[2] S. Bloch, “An example in the theory of algebraic cycles”, Algebraic K-theory (Northwestern Univ., Evanston, IL, 1976), Lecture Notes in Math., 551, Springer, Berlin, 1976, 1–29 | DOI | MR | Zbl

[3] S. Bloch, “Torsion algebraic cycles and a theorem of Roitman”, Compositio Math., 39:1 (1979), 107–127 | MR | Zbl

[4] S. Bloch, J. P. Murre, “On the Chow group of certain types of Fano threefolds”, Compositio Math., 39:1 (1979), 47–105 | MR | Zbl

[5] J.-L. Colliot-Thélène, J.-J. Sansuc, C. Soulé, “Torsion dans le groupe de Chow de codimension deux”, Duke Math. J., 50:3 (1983), 763–801 | DOI | MR | Zbl

[6] O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001, xiv+233 pp. | DOI | MR | Zbl

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

[8] P. Deligne, “La conjecture de Weil. II”, Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252 | DOI | MR | Zbl

[9] G. Faltings, “Complements to Mordell”, Rational points (Bonn, 1983/1984), Aspects Math., E6, Friedr. Viehweg, Braunschweig, 1984, 203–227 | MR | Zbl

[10] E. Freitag, R. Kiehl, Étale cohomology and the Weil conjecture, transl. from the German, with an historical introduction by J. A. Dieudonné, Ergeb. Math. Grenzgeb. (3), 13, Springer-Verlag, Berlin, 1988, xviii+317 pp. | DOI | MR | Zbl

[11] S. Gorchinskiy, V. Guletskiĭ, “Motives and representability of algebraic cycles on threefolds over a field”, J. Algebraic Geom., 21:2 (2012), 347–373 | DOI | MR | Zbl

[12] P. Deligne, N. Katz, Groupes de monodromie en géométrie algébrique, Séminaire de géométrie algébrique du Bois-Marie 1967–1969 (SGA 7 II), v. II, Lecture Notes in Math., 340, Springer-Verlag, Berlin–New York, 1973, x+438 pp. | DOI | MR | Zbl

[13] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp. | DOI | MR | Zbl

[14] K. Lamotke, “The topology of complex projective varieties after S. Lefschetz”, Topology, 20:1 (1981), 15–51 | DOI | MR | Zbl

[15] A. S. Merkur'ev, A. A. Suslin, “$K$-cohomology of Severi–Brauer varieties and the norm residue homomorphism”, Math. USSR-Izv., 21:2 (1983), 307–340 | DOI | MR | Zbl

[16] D. Mumford, “Rational equivalence of $0$-cycles on surfaces”, J. Math. Kyoto Univ., 9:2 (1969), 195–204 | DOI | MR | Zbl | Zbl

[17] J. P. Murre, “Un résultat en théorie des cycles algébriques de codimension deux”, C. R. Acad. Sci. Paris Sér. I Math., 296:23 (1983), 981–984 | MR | Zbl

[18] A. A. Roĭtman, “On $\Gamma$-equivalence of zero-dimensional cycles”, Math. USSR-Sb., 15:4 (1971), 555–567 | DOI | MR | Zbl

[19] A. A. Rojtman, “The torsion of the group of $0$-cycles modulo rational equivalence”, Ann. of Math. (2), 111:3 (1980), 553–569 | DOI | MR | Zbl

[20] C. Schoen, “On Hodge structures and non-representability of Chow groups”, Compositio Math., 88:3 (1993), 285–316 | MR | Zbl

[21] S. S. Shatz, “Group schemes, formal groups, and $p$-divisible groups”, Arithemtic geometry (Storrs, CT, 1984), Springer, New York, 1986, 29–78 | MR | Zbl

[22] Mingmin Shen, Surfaces with involution and Prym constructions, arXiv: 1209.5457v2

[23] Mingmin Shen, “On relations among $1$-cycles on cubic hypersurfaces”, J. Algebraic Geom., 23:3 (2014), 539–569 | DOI | MR | Zbl

[24] A. Suslin, V. Voevodsky, “Relative cycles and Chow sheaves”, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000, 10–86 | MR | Zbl

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

[26] C. Vial, “Algebraic cycles and fibrations”, Doc. Math., 18 (2013), 1521–1553 | MR | Zbl

[27] C. Voisin, “Théorème de Torelli pour les cubiques de $\mathbb P^5$”, Invent. Math., 86:3 (1986), 577–601 | DOI | MR | Zbl

[28] C. Voisin, “Sur les zéro-cycles de certaines hypersurfaces munies d'un automorphisme”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19:4 (1992), 473–492 | MR | Zbl

[29] C. Voisin, “Variations de structure de Hodge et zéro-cycles sur les surfaces générales”, Math. Ann., 299:1 (1994), 77–103 | DOI | MR | Zbl

[30] C. Voisin, Hodge theory and complex algebraic geometry, transl. from the French, v. I, Cambridge Stud. Adv. Math., 76, Cambridge Univ. Press, Cambridge, 2002, x+322 pp. ; v. II, Cambridge Stud. Adv. Math., 77, 2003, x+351 pp. | DOI | MR | Zbl | DOI | MR | Zbl

[31] C. Voisin, “Symplectic involutions of $K3$-surfaces act trivially on $\mathrm{CH}_0$”, Doc. Math., 17 (2012), 851–860 | MR | Zbl

[32] C. Voisin, “The generalized Hodge and Bloch conjectures are equivalent for general complete intersections”, Ann. Sci. Éc. Norm. Supér. (4), 46:3 (2013), 449–475 | DOI | MR | Zbl

[33] C. Voisin, “On the universal $\mathrm{CH}_0$ of cubic hypersurfaces”, J. Eur. Math. Soc. (JEMS), 19:6 (2017), 1619–1653 | DOI | MR | Zbl