## Rocky Mountain Journal of Mathematics

### On Chow groups of some hyperkahler fourfolds with a non-symplectic involution, II

Robert Laterveer

#### Abstract

This note is about hyperkahler 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 $X/\iota$.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1925-1950.

Dates
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/euclid.rmjm/1543028446

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1925

Mathematical Reviews number (MathSciNet)
MR3879310

Zentralblatt MATH identifier
06987233

#### Citation

Laterveer, Robert. On Chow groups of some hyperkahler fourfolds with a non-symplectic involution, II. Rocky Mountain J. Math. 48 (2018), no. 6, 1925--1950. doi:10.1216/RMJ-2018-48-6-1925. https://projecteuclid.org/euclid.rmjm/1543028446

#### References

• A. Beauville, Some remarks on Kähler manifolds with $c_1=0$, in: Classification of algebraic and analytic manifolds, Birkhäuser, Boston, 1983.
• ––––, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differ. Geom. 18 (1983), 755–782.
• ––––, On the splitting of the Bloch-Beilinson filtration, in: Algebraic cycles and motives, J. Nagel and C. Peters, eds., Cambridge University Press, Cambridge, 2007.
• 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.
• C. Camere, Symplectic involutions of holomorphic symplectic fourfolds, Bull. Lond. Math. Soc. 44 (2012), 687–702.
• C. Delorme, Espaces projectifs anisotropes, Bull. Soc. Math. France 103 (1975), 203–223.
• L. Fu, Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections, Adv. Math. 244 (2013), 894–924.
• ––––, On the action of symplectic automorphisms on the $CH_0$-groups of some hyper-Kähler fourfolds, Math. Z. 280 (2015), 307–334.
• ––––, Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds, Glasgow Math. J. 58 (2016), 17–37.
• L. Fu, R. Laterveer and C. Vial, The generalized Franchetta conjecture for some hyper–Kähler varieties, arXiv:1708.02919.
• L. Fu, Z. Tian and C. Vial, Motivic hyperkähler resolution conjecture for generalized Kummer varieties, arXiv:1608.04968.
• W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.
• U. Jannsen, Motivic sheaves and filtrations on Chow groups, in Motives, U. Jannsen, et al., eds., Proc. Symp. Pure Math. 55 (1994).
• R. Laterveer, Algebraic cycles on a very special EPW sextic, Rend. Sem. Mat. Univ. Padova, to appear.
• ––––, About Chow groups of certain hyperkähler varieties with non-symplectic automorphisms, Vietnam J. Math. 46 (2018), 453–470.
• ––––, On the Chow groups of some hyperkähler fourfolds with a non-symplectic involution, Inter. J. Math. 28 (2017), 1–18.
• ––––, On the Chow groups of certain EPW sextics, submitted.
• J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, Parts I and II, Indag. Math. 4 (1993), 177–201.
• J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. Univ. Lect. Ser. 61, Providence, 2013.
• K. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. Math. 139 (1994), 641–660.
• T. Scholl, Classical motives, in Motives, U. Jannsen, et al., eds., Proc. Symp. Pure Math. 55 (1994).
• M. Shen and C. Vial, The Fourier transform for certain hyperKähler fourfolds, Mem. Amer. Math. Soc. 240 (2016).
• ––––, The motive of the Hilbert cube $X^{[3]}$, Forum Math. Sigma 4 (2016).
• C. Vial, Remarks on motives of abelian type, Tohoku Math. J. 69 (2017), 195–220.
• ––––, On the motive of some hyperkähler varieties, J. reine angew. Math. 725 (2017), 235–247.
• C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spec. Soc. Math. France, Paris, 2002.
• ––––, Chow rings and decomposition theorems for $K3$ surfaces and Calabi-Yau hypersurfaces, Geom. Topol. 16 (2012), 433–473.
• ––––, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, Ann. Sci. Ecole Norm. Sup. 46 (2013), 449–475.
• ––––, Bloch's conjecture for Catanese and Barlow surfaces, J. Differ. Geom. 97 (2014), 149–175.
• ––––, Chow rings, Decomposition of the diagonal, and the topology of families, Princeton University Press, Princeton, 2014.
• ––––, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II, J. Math. Sci. Univ. Tokyo 22 (2015), 491–517.