2024/04/16 by Laterveer, Robert, Vial, Charles
#14C15 #14C25 #14C30 #14J28 #14J42 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.10465
The Chow rings of hyper-Kähler varieties are conjectured to have a particularly rich structure. In this paper, we formulate a conjecture that combines the Beauville-Voisin conjecture regarding the subring generated by divisors and the Franchetta conjecture regarding generically defined cycles. As motivation, we show that this Beauville-Voisin-Franchetta conjecture for a hyper-Kähler variety X follows from a combination of Grothendieck's standard conjectures for a very general deformation of X, Murre's conjecture (D) for X and the Franchetta conjecture for X3. As evidence, beyond the case of Fano varieties of lines on smooth cubic fourfolds, we show that this conjecture holds for codimension-2 and codimension-8 cycles on Lehn-Lehn-Sorger-van Straten eightfolds. Moreover, we establish that the subring of the Chow ring generated by primitive divisors injects into cohomology.