Abstract
We prove the generalized Franchetta conjecture for the locally complete family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr\`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.
Highlights
The Franchetta property. — Let f : X → B be a smooth projective morphism between smooth schemes of finite type over the field of complex numbers
By adapting and refining an argument of Bülles [Bül20], we show in Theorem 1.1 that the motive of M belongs to the thick subcategory generated by Tate twists of the motive of Y m, where dim M = 2m
Proposition 2.11 below, which parallels [Tav18] in the case of hyperelliptic curves and [Yin15a] in the case of K3 surfaces, shows that for a Fano or Calabi–Yau hypersurface Y the only non-trivial relations among tautological cycles in powers of Y are given by the multiplicative Chow–Künneth (MCK) relation (11) and the finite-dimensionality relation
Summary
The Franchetta property. — Let f : X → B be a smooth projective morphism between smooth schemes of finite type over the field of complex numbers. The generalized Franchetta conjecture for some hyper-Kähler varieties, II in the case m = 2 in [FLVS19] for the universal family of K3 surfaces of genus 12 (but different from 11) and for the Beauville–Donagi family of Fano varieties of lines on smooth cubic fourfolds. — The universal family of LLSS hyper-Kähler eightfolds over the moduli space of smooth cubic fourfolds not containing a plane satisfies the Franchetta property. One further obtains as a direct consequence of Theorem 1 that for an LLSS eightfold Z, the subring of CH∗(Z) generated by the polarization h, the Chern classes cj(Z) and the classes of the (generically defined) co-isotropic subvarieties described in [FLVS19, Cor. 1.12] injects in cohomology via the cycle class map This provides new evidence for Voisin’s refinement in [Voi16] of the Beauville–Voisin conjecture. We thank the referees for constructive comments that improved our paper
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