A family of cubic fourfolds with finite-dimensional motive. (English) Zbl 1408.14019

Laterveer, Robert

J. Math. Soc. Japan 70, No. 4, 1453-1473 (2018).

The paper considers smooth cubic fourfolds \(X\) which are cyclic covers of \(\mathbb P^4\) branched along a smooth cubic threefold, so their equation is \(x^3_5 + f(x_0, \dots, x_4) = 0 \). The main object is to prove that such fourfolds have finite-dimensional Chow motive, in Kimura’s sense [S.-I. Kimura, Math. Ann. 331, No. 1, 173–201 (2005; Zbl 1067.14006)]. As the author explains, the strategy is to exploit a theorem of B. van Geemen and E. Izadi [Math. Z. 242, No. 2, 279–301 (2002; Zbl 1050.14009)], which yields the existence of a correspondence \(\Gamma\) in \(A^5 (X \times Z \times E)\) such that it determines an embedding \(H^4 (X)_{\mathrm{prim}} \hookrightarrow H^5(Z) \otimes H^1(E),\) here \(Z\) is a cubic fivefold and \(E\) is the Fermat elliptic curve. It follows that the homological motive of \(X\) is of abelian type, the issue then is to show that this is true already for rational equivalence. The idea is to check that \(\Gamma\) comes from the restriction of a correspondence in the universal family. To this aim, Voisin’s method of spread is applied and then the main conclusions is reached by modifying some ideas from a result of L. Fu [Math. Z. 280, No. 1–2, 307–334 (2015; Zbl 1388.14121)].

Reviewer: Alberto Collino (Verzuolo)

MSC:

14C15	(Equivariant) Chow groups and rings; motives
14C25	Algebraic cycles
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14K99	Abelian varieties and schemes

Keywords:

motives; cubic fourfolds; abelian varieties; algebraic cycles; Chow groups; finite-dimensional motives; Kuga-Satake correspondence

Citations:

Zbl 1067.14006; Zbl 1050.14009; Zbl 1388.14121

References:

[1]	J. Achter, S. Casalaina-Martin and C. Vial, On descending cohomology geometrically, to appear in Comp. Math., arXiv:1410.5376. · Zbl 1370.14039
[2]	Y. André, Motifs de dimension finie (d’après S.-I. Kimura, P. O’Sullivan,...), Séminaire Bourbaki 2003/2004, Astérisque 299, Exp. No. 929, viii, 115–145.
[3]	A. Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann., 273 (1986), 647–651. · Zbl 0566.14003
[4]	S. Bloch, Lectures on algebraic cycles, Duke Univ. Press Durham, 1980. · Zbl 0436.14003
[5]	S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup., 4 (1974), 181–202. · Zbl 0307.14008
[6]	S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math., 105 (1983), 1235–1253. · Zbl 0525.14003
[7]	S. Boissière, C. Camere and A. Sarti, Classification of automorphisms on a deformation family of hyperkähler fourfolds by \(p\)-elementary lattices, Kyoto J. Math., 56 (2016), 465–499. · Zbl 1375.14143
[8]	M. Brion, Log homogeneous varieties, In: Actas del XVI Coloquio Latinoamericano de Algebra, Revista Matemática Iberoamericana, Madrid 2007..
[9]	M. de Cataldo and L. Migliorini, The Chow groups and the motive of the Hilbert scheme of points on a surface, J. Algebra, 251 (2002), 824–848. · Zbl 1033.14004
[10]	A. Collino, The Abel–Jacobi isomorphism for the cubic fivefold, Pacific J. Math., 122 (1986), 43–55. · Zbl 0618.14018
[11]	P. Deligne, La conjecture de Weil pour les surfaces \(K3\), Invent. Math., 15 (1972), 206–226. · Zbl 0219.14022
[12]	C. Delorme, Espaces projectifs anisotropes, Bull. Soc. Math. France, 103 (1975), 203–223. · Zbl 0314.14016
[13]	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
[14]	I. Dolgachev, Weighted projective varieties, In: Group actions and vector fields, Vancouver, 1981, Springer Lecture Notes in Math., 956, Springer Berlin Heidelberg New York, 1982.
[15]	L. Fu, On the action of symplectic automorphisms on the \(CH_0\)-groups of some hyper-Kähler fourfolds, Math. Z., 280 (2015), 307–334. · Zbl 1388.14121
[16]	W. Fulton, Intersection theory, Springer-Verlag Ergebnisse der Mathematik, Berlin Heidelberg New York Tokyo, 1984, · Zbl 0541.14005
[17]	B. van Geemen, Half twists of Hodge structures of CM-type, J. Math. Soc. Japan, 53 (2001), 813–833. · Zbl 1074.14509
[18]	B. van Geemen and E. Izadi, Half twists and the cohomology of hypersurfaces, Math. Z., 242 (2002), 279–301. · Zbl 1050.14009
[19]	A. Hirschowitz and J. Iyer, Hilbert schemes of fat r-planes and the triviality of Chow groups of complete intersections, In: Vector bundles and complex geometry, Contemp. Math., 522, Amer. Math. Soc., Providence, 2010. · Zbl 1219.14009
[20]	J. Iyer, Murre’s conjectures and explicit Chow–Künneth projectors for varieties with a nef tangent bundle, Transactions of the Amer. Math. Soc., 361 (2008), 1667–1681.
[21]	J. Iyer, Absolute Chow–Künneth decomposition for rational homogeneous bundles and for log homogeneous varieties, Michigan Math. J., 60 (2011), 79–91. · Zbl 1233.14003
[22]	U. Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math., 107 (1992), 447–452. · Zbl 0762.14003
[23]	U. Jannsen, On finite-dimensional motives and Murre’s conjecture, In: Algebraic cycles and motives (eds. J. Nagel and C. Peters), Cambridge University Press, Cambridge, 2007. · Zbl 1127.14007
[24]	B. Kahn, J. Murre and C. Pedrini, On the transcendental part of the motive of a surface, In: Algebraic cycles and motives (eds. J. Nagel and C. Peters), Cambridge University Press, Cambridge, 2007. · Zbl 1130.14008
[25]	B. Kahn and R. Sebastian, Smash-nilpotent cycles on abelian 3-folds, Math. Res. Letters, 16 (2009), 1007–1010. · Zbl 1191.14009
[26]	S. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann., 331 (2005), 173–201. · Zbl 1067.14006
[27]	S. Kleiman, The standard conjectures, In: Motives (eds. U. Jannsen et alii), Proceedings of Symposia in Pure Mathematics, 55 (1994), Part 1. · Zbl 0820.14006
[28]	K. Künnemann, A Lefschetz decomposition for Chow motives of abelian schemes, Inv. Math., 113 (1993), 85–102. · Zbl 0806.14001
[29]	R. Laterveer, Algebraic cycles and Todorov surfaces, to appear in Kyoto J. Math., arXiv:1609.09629. · Zbl 1402.14007
[30]	R. Laterveer, A remark on the motive of the Fano variety of lines of a cubic, Ann. Math. Québec, 41 (2017), 141–154. · Zbl 1386.14027
[31]	R. Laterveer, Some new examples of smash-nilpotent algebraic cycles, Glasgow Math. J., 59 (2017), 623–634. · Zbl 1378.14005
[32]	J. Lewis, Cylinder homomorphisms and Chow groups, Math. Nachr., 160 (1993), 205–221. · Zbl 0802.14003
[33]	J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag. Math., 4 (1993), 177–201. · Zbl 0805.14001
[34]	J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. University Lecture Series, 61, Providence, 2013. · Zbl 1273.14002
[35]	A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degré, C. R. Acad. Sci. Paris Série I Math., 329 (1999), 51–56. · Zbl 0981.14004
[36]	K. Paranjape, Abelian varieties associated to certain K3 surfaces, Comp. Math., 68 (1988), 11–22. · Zbl 0698.14037
[37]	C. Pedrini, On the finite dimensionality of a \(K3\) surface, Manuscripta Mathematica, 138 (2012), 59–72. · Zbl 1278.14012
[38]	C. Pedrini, Bloch’s conjecture and valences of correspondences for \(K3\) surfaces, arXiv:1510.05832v1.
[39]	C. Pedrini, On the rationality and the finite dimensionality of a cubic fourfold, arXiv:1701.05743. · Zbl 1278.14012
[40]	U. Rieß, On the Chow ring of birational irreducible symplectic varieties, Manuscripta Math., 145 (2014), 473–501. · Zbl 1325.14016
[41]	T. Scholl, Classical motives, In: Motives (eds. U. Jannsen et alii), Proceedings of Symposia in Pure Math., 55 (1994), Part 1. · Zbl 0814.14001
[42]	R. Sebastian, Smash nilpotent cycles on varieties dominated by products of curves, Comp. Math., 149 (2013), 1511–1518. · Zbl 1285.14003
[43]	R. Sebastian, Examples of smash nilpotent cycles on rationally connected varieties, J. Algebra, 438 (2015), 119–129. · Zbl 1323.14007
[44]	T. Shioda, The Hodge conjecture for Fermat varieties, Math. Ann., 245 (1979), 175–184. · Zbl 0403.14007
[45]	C. Vial, Algebraic cycles and fibrations, Documenta Math., 18 (2013), 1521–1553. · Zbl 1349.14027
[46]	C. Vial, Projectors on the intermediate algebraic Jacobians, New York J. Math., 19 (2013), 793–822. · Zbl 1292.14005
[47]	C. Vial, Remarks on motives of abelian type, Tohoku Math. J., 69 (2017), 195–220. · Zbl 1386.14031
[48]	C. Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proceedings of the LMS, 106 (2013), 410–444. · Zbl 1271.14010
[49]	C. Vial, Chow–Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles, J. Alg. Geom., 24 (2015), 51–80. · Zbl 1323.14006
[50]	V. Voevodsky, A nilpotence theorem for cycles algebraically equivalent to zero, Internat. Math. Research Notices, 4 (1995), 187–198. · Zbl 0861.14006
[51]	C. Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, Ann. Sci. Ecole Norm. Sup., 46 (2013), fascicule 3, 449–475. · Zbl 1282.14015
[52]	C. Voisin, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geometry, 97 (2014), 149–175. · Zbl 1386.14145
[53]	C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton University Press, Princeton and Oxford, 2014. · Zbl 1288.14001
[54]	C. Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II, J. Math. Sci. Univ. Tokyo, 22 (2015), 491–517. · Zbl 1332.14014
[55]	Z. Xu, Algebraic cycles on a generalized Kummer variety, arXiv:1506.04297v1.