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Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for abelian varieties. (English) Zbl 1486.14014

The paper under review studies classical questions on Chow groups of abelian varieties and gives some application to Chow groups of varieties “coming from” abelian varieties such as generalized Kummer varieties. The questions are the generalized Hodge conjecture and the generalized Bloch conjecture; hence both the image and the kernel of the class cycle map are studied. Chow groups are considered with rational coefficients and the paper focuses on the \(\mathbb{Q}\)-sub-algebra \(R^*(A)\subset CH^*(A)\) generated by divisors.
There is a long list of works on the subject and it is remarkable that the author can say something new. The first who studied \(R^*(A)\) are probably F. Hazama [J. Fac. Sci., Univ. Tokyo, Sect. I A 31, 487–520 (1984; Zbl 0591.14006)] and V. K. Murty [Math. Ann. 268, 197–206 (1984; Zbl 0521.14004)] who described the image of the class cycle map to singular cohomology \[ cl : R^*(A) \longrightarrow H^*(A). \] One can define the Lefschetz group of \(A\), \(L(A)\subset GL(H^1(A))\), as the subgroup of elements of \(GL(H^1(A))\) commuting with the action of the endomorphisms of \(A\) and respecting all pairings induced by polarizations. Notice that \(L(A)\) acts on \(H^n(A)\) through the identification \(H^n(A)= \Lambda^n H^1(A).\) With the above notation, Hazama and Murty showed the equality \[ cl( R^*(A) ) = H^*(A)^{L(A)}. \] The point to show such an equality is to prove that the invariants of \(L(A)\) are generated in degree two. This asks to analyse some cases. Depending on the endomorphisms of \(A\), following Albert’s classification, \(L(A) \times_\mathbb{Q} \bar{\mathbb{Q}}\) is the product of some linear groups, orthogonal groups or symplectic groups. It turns out that, by the first fundamental theorem of invariant theory, the invariants of all these three classical groups are generated in degree two.
The second fundamental theorem of invariant theory, describes the relation between these invariants. Base on it, it is possible to the describe the kernel \(\ker_{\vert R^*(A) }\). For details see the PhD thesis of the reviewer.
The results of Hazama and Murty give some evidences on the Hodge conjecture, as in some cases the Lefschetz group coincide with the Mumford-Tate group. Similarly, the description of \(\ker_{\vert R^*(A) }\) we mentioned before corresponds to the Bloch conjecture for sub-algebra \(R^*(A)\subset CH^*(A)\). The paper under review extends this study and focuses on the generalized Hodge and Bloch conjecture. From a group theoretic point of view this amounts to passing from the study of the invariants of \(L(A) \) and their relation to the study of the sub-\(L(A) \)-representation and the relations between their endomorphisms. Some technical complication appear which force the author to exclude some cases of Albert’s classification from his main results.

MSC:

14C25 Algebraic cycles
14C15 (Equivariant) Chow groups and rings; motives
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K10 Algebraic moduli of abelian varieties, classification
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[1] S. ABDULALI, Hodge structures of CM-type, J. Ramanujan Math. Soc. 20 (2005), 155-162. · Zbl 1094.14004
[2] S. ABDULALI, Tate twists of Hodge structures arising from Abelian varieties of type IV, J. Pure Appl. Algebra 216 (2012), 1164-1170. · Zbl 1247.14007
[3] S. ABDULALI, Tate twists of Hodge structures arising from Abelian varieties, In: “Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles, and Arithmetic” M. Kerr and G. Pearlstein (eds.), London Mathematical Society Lecture Note Series Vol. 427, Cam-bridge University Press, 2016, 292-307. · Zbl 1360.14024
[4] J. ACHTER, S. CASALAINA-MARTIN and C. VIAL, Distinguished models of intermediate Jacobians, J. Inst. Math. Jussieu 19 (2020), 891-918. · Zbl 1446.14029
[5] G. ANCONA, “Décomposition du motif d”un schéma abélien universel”, Ph.D. thesis, Uni-versité Paris, Thèse de doctorat Mathématiques, 2012.
[6] G. ANCONA, Décomposition de motifs abéliens, Manuscripta Math. 146 (2015), 307-328. · Zbl 1453.14017
[7] Y. ANDRÉ, Pour une théorie inconditionnelle des motifs, Inst. HautesÉtudes Sci. Publ. Math. 83 (1996), 5-49. · Zbl 0874.14010
[8] Y. ANDRÉ, “Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes)”, Panora-mas et Synthèses Vol. 17, Société Mathématique de France, Paris, 2004. · Zbl 1060.14001
[9] A. BEAUVILLE, Variétés Kählériennes dont la première classe de Chern est nulle, J. Dif-ferential Geom. 18 (1983), 755-782. · Zbl 0537.53056
[10] A. BEAUVILLE, Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, In: “Algebraic geometry (Tokyo/Kyoto, 1982)”, Lecture Notes in Math., Vol. 1016, Springer, Berlin, 1983, 238-260. · Zbl 0526.14001
[11] A. BEAUVILLE, Sur l’anneau de Chow d’une variété abélienne, Math. Ann. 273 (1986), 647-651. · Zbl 0566.14003
[12] C. BIRKENHAKE and H. LANGE, The dual polarization of an Abelian variety, Arch. Math. (Basel) 73 (1999), 380-389. · Zbl 0979.14024
[13] S. BLOCH, A. KAS and D. LIEBERMAN, Zero cycles on surfaces with p g = 0, Compos. Math. 33 (1976), 135-145. · Zbl 0337.14006
[14] S. BLOCH and V. SRINIVAS, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), 1235-1253. · Zbl 0525.14003
[15] N. BOURBAKI, “Groupes et Algèbres de Lie”, Chapitres VII et VIII, Hermann, Paris 1975. · Zbl 0329.17002
[16] G. CERESA, C is not algebraically equivalent to C in its Jacobian, Ann. of Math. 117 (1983), 285-291. · Zbl 0538.14024
[17] C. DENINGER and J. MURRE, Motivic decomposition of Abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201-219. · Zbl 0745.14003
[18] E. FRIEDLANDER, Filtrations on algebraic cycles and homology, Ann. Sci.Éc. Norm. Supér. 28 (1995), 317-343. · Zbl 0854.14006
[19] R. FRINGUELLI and R. PIRISI, The Picard group of the universal Abelian variety and the Franchetta conjecture for Abelian varieties, Michigan Math. J. 68 (2019), 651-671. · Zbl 1430.14019
[20] L. FU, On the action of symplectic automorphisms on the C H 0 -groups of some hyper-Kähler fourfolds, Math. Z. 280 (2015), 307-334. · Zbl 1388.14121
[21] L. FU, R. LATERVEER and C. VIAL, The generalized Franchetta conjecture for some hyper-Kaehler varieties, J. Math. Pures Appl. 130 (2019), 1-35. · Zbl 1423.14033
[22] L. FU, Z. TIAN and C. VIAL, Motivic hyperKähler resolution conjecture: I. Generalized Kummer varieties, Geom. Topol. 23 (2019), 427-492. · Zbl 1520.14009
[23] L. FU and C. VIAL, Distinguished cycles on varieties with motive of Abelian type and the section property, J. Algebraic Geom. 29 (2020), 53-107. · Zbl 1430.14020
[24] W. FULTON and J. HARRIS, “Representation Theory”, A first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991, xvi+551. · Zbl 0744.22001
[25] F. HAZAMA, Algebraic cycles on certain Abelian varieties and powers of special surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), 487-520. · Zbl 0591.14006
[26] F. HAZAMA, The generalized Hodge conjecture for stably nondegenerate Abelian varieties, Compos. Math. 93 (1994), 129-137. · Zbl 0848.14003
[27] F. HAZAMA, On the general Hodge conjecture for Abelian varieties of CM-type, Publ. Res. Inst. Math. Sci. 39 (2003), 625-655. · Zbl 1067.14010
[28] D. HUYBRECHTS, Symplectic automorphisms of K3 surfaces of arbritrary finite order, Math. Res. Lett. 19 (2012), 947-951. · Zbl 1266.14030
[29] S.-I. KIMURA, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173-201. · Zbl 1067.14006
[30] S. KLEIMAN, Algebraic cycles and the Weil conjectures, In: “Dix Exposés sur la Coho-mologie des Schémas”, Adv. Stud. Pure Math., Vol. 3, North-Holland, Amsterdam, 1968, 359-386. · Zbl 0198.25902
[31] R. LATERVEER, Some results on a conjecture of Voisin for surfaces of geometric genus one, Boll. Unione Mat. Ital. 9 (2016), 435-452. · Zbl 1375.14025
[32] R. LATERVEER, Some desultory remarks concerning algebraic cycles and Calabi-Yau threefolds, Rend. Circ. Mat. Palermo (2) 65 (2016), 333-344. · Zbl 1360.14017
[33] H.-Y. LIN, On the Chow group of zero-cycles of a generalized Kummer variety, Adv. Math. 298 (2016), 448-472. · Zbl 1343.14005
[34] H.-Y. LIN, Corrigendum to On the Chow group of zero-cycles of a generalized Kummer variety, Adv. Math. 331 (2018), 1016-1021. · Zbl 1439.14029
[35] J. MILNE, Lefschetz classes on Abelian varieties, Duke Math. J. 96 (1999), 639-675. · Zbl 0976.14009
[36] B. MOONEN and Y. ZARHIN, Hodge classes on Abelian varieties of low dimension, Math. Ann. 315 (1999), 711-733. · Zbl 0947.14005
[37] B. MOONEN, On the Chow motive of an Abelian scheme with non-trivial endomorphisms, J. Reine Angew. Math. 711 (2016), 75-109. · Zbl 1342.14010
[38] D. MUMFORD, J. FOGARTY and F. KIRWAN, “Geometric Invariant Theory”, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 34, Springer-Verlag, Berlin, 1994. · Zbl 0797.14004
[39] K. MURTY, Exceptional Hodge classes on certain Abelian varieties, Math. Ann. 268 (1984), 197-206. · Zbl 0521.14004
[40] M. NORI, Cycles on the generic Abelian threefold, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), 191-196. · Zbl 0725.14006
[41] P. O’SULLIVAN, Algebraic cycles on an Abelian variety, J. Reine Angew. Math. 654 (2011), 1-81. · Zbl 1258.14006
[42] R. PAWAR, Action of correspondences on filtrations on cohomology and 0-cycles of Abelian varieties, Math. Z. 292 (2019), 655-675. · Zbl 1448.14012
[43] K. RIBET, Hodge classes on certain types of Abelian varieties, Amer. J. Math. 105 (1983), 523-538. · Zbl 0586.14003
[44] M. SHEN and C. VIAL, “The Fourier Transform for Certain HyperKähler Fourfolds”, Mem. Amer. Math. Soc., Vol. 240, 2016. · Zbl 1386.14025
[45] A. SHERMENEV, Motif of an Abelian variety, Funckcional. Anal. i Priložen 8 (1974), 55-61. · Zbl 0294.14003
[46] G. SHIMURA, On analytic families of polarized Abelian varieties and automorphic func-tions, Ann. of Math. 78 (1963), 149-192. · Zbl 0142.05402
[47] T. SCHOLL, Classical motives, In: “Motives (Seattle, WA, 1991)”, Proc. Sympos. Pure Math., Vol. 55, 1994, 163-187. · Zbl 0814.14001
[48] S. G. TANKEEV, Cycles on simple Abelian varieties of prime dimension, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 155-170, 192.
[49] S. G. TANKEEV, Abelian varieties and the general Hodge conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), 192-206. · Zbl 0871.14009
[50] C. VIAL, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. Lond. Math. Soc. 106 (2013), 410-444. · Zbl 1271.14010
[51] V. VOEVODSKY, A nilpotence theorem for cycles algebraically equivalent to zero, Int. Math. Res. Not. IMRN (1995), 187-198. · Zbl 0861.14006
[52] C. VOISIN, Sur les zéro-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 473-492. · Zbl 0786.14006
[53] C. VOISIN, Remarks on zero-cycles of self-products of varieties, In: “Moduli of Vector Bundles” (Sanda, 1994; Kyoto, 1994), 265-285, Lecture Notes in Pure and Appl. Math., Vol. 179, Dekker, New York, 1996. · Zbl 0912.14003
[54] C. VOISIN, Symplectic involutions of K3 surfaces act trivially on C H 0 , Doc. Math. 17 (2012), 851-860. · Zbl 1276.14012
[55] C. VOISIN, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geom. 97 (2014), 149-175. · Zbl 1386.14145
[56] C. VOISIN, “Chow Rings, Decomposition of the Diagonal, and the Topology of Families”, Annals of Mathematics Studies, Vol. 187, Princeton University Press, Princeton, NJ, 2014. · Zbl 1288.14001
[57] Z. XU, Algebraic cycles on a generalized Kummer variety, Int. Math. Res. Not. IMRN (2018), 932-948. · Zbl 1423.14057
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