# zbMATH — the first resource for mathematics

On Chow groups of some hyperkähler fourfolds with a non-symplectic involution. II. (English) Zbl 1402.14008
Summary: This note is about hyperkähler fourfolds $$X$$ admitting a non-symplectic involution $$\iota$$. The Bloch-Beilinson conjectures predict the way $$\iota$$ should act on certain pieces of the Chow groups of $$X$$. The main result of this note is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $$X/\iota$$.
For Part I see [the author, Int. J. Math. 28, No. 3, Article ID 1750014, 19 p. (2017; Zbl 1397.14010)].
##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14C25 Algebraic cycles 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
Full Text:
##### References:
 [1] A. Beauville, Some remarks on Kähler manifolds with $$c_1=0$$, in: Classification of algebraic and analytic manifolds, Birkhäuser, Boston, 1983. [2] ——–, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom. 18 (1983), 755–782. [3] ——–, On the splitting of the Bloch-Beilinson filtration, in: Algebraic cycles and motives, J. Nagel and C. Peters, eds., Cambridge University Press, Cambridge, 2007. [4] A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension $$4$$, C.R. Acad. Sci. Paris 301 (1985), 703–706. [5] C. Camere, Symplectic involutions of holomorphic symplectic fourfolds, Bull. Lond. Math. Soc. 44 (2012), 687–702. · Zbl 1254.14050 [6] C. Delorme, Espaces projectifs anisotropes, Bull. Soc. Math. France 103 (1975), 203–223. · Zbl 0314.14016 [7] L. Fu, Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections, Adv. Math. 244 (2013), 894–924. · Zbl 1300.14007 [8] ——–, On the action of symplectic automorphisms on the $$CH_0$$-groups of some hyper-Kähler fourfolds, Math. Z. 280 (2015), 307–334. [9] ——–, Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds, Glasgow Math. J. 58 (2016), 17–37. · Zbl 1393.14042 [10] L. Fu, R. Laterveer and C. Vial, The generalized Franchetta conjecture for some hyper–Kähler varieties, arXiv:1708.02919. [11] L. Fu, Z. Tian and C. Vial, Motivic hyperkähler resolution conjecture for generalized Kummer varieties, arXiv:1608.04968. [12] W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005 [13] U. Jannsen, Motivic sheaves and filtrations on Chow groups, in Motives, U. Jannsen, et al., eds., Proc. Symp. Pure Math. 55 (1994). · Zbl 0811.14004 [14] R. Laterveer, Algebraic cycles on a very special EPW sextic, Rend. Sem. Mat. Univ. Padova, to appear. · Zbl 1422.14013 [15] ——–, About Chow groups of certain hyperkähler varieties with non-symplectic automorphisms, Vietnam J. Math. 46 (2018), 453–470. [16] ——–, On the Chow groups of some hyperkähler fourfolds with a non-symplectic involution, Inter. J. Math. 28 (2017), 1–18. [17] ——–, On the Chow groups of certain EPW sextics, submitted. [18] 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 [19] J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. Univ. Lect. Ser. 61, Providence, 2013. · Zbl 1273.14002 [20] K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. Math. 139 (1994), 641–660. · Zbl 0828.14003 [21] T. Scholl, Classical motives, in Motives, U. Jannsen, et al., eds., Proc. Symp. Pure Math. 55 (1994). · Zbl 0814.14001 [22] M. Shen and C. Vial, The Fourier transform for certain hyperKähler fourfolds, Mem. Amer. Math. Soc. 240 (2016). · Zbl 1386.14025 [23] ——–, The motive of the Hilbert cube $$X^{[3]}$$, Forum Math. Sigma 4 (2016). · Zbl 1362.14003 [24] C. Vial, Remarks on motives of abelian type, Tohoku Math. J. 69 (2017), 195–220. · Zbl 1386.14031 [25] ——–, On the motive of some hyperkähler varieties, J. reine angew. Math. 725 (2017), 235–247. [26] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spec. Soc. Math. France, Paris, 2002. [27] ——–, Chow rings and decomposition theorems for $$K3$$ surfaces and Calabi-Yau hypersurfaces, Geom. Topol. 16 (2012), 433–473. · Zbl 1253.14005 [28] ——–, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, Ann. Sci. Ecole Norm. Sup. 46 (2013), 449–475. · Zbl 1282.14015 [29] ——–, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differ. Geom. 97 (2014), 149–175. · Zbl 1386.14145 [30] ——–, Chow rings, Decomposition of the diagonal, and the topology of families, Princeton University Press, Princeton, 2014. [31] ——–, 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.