The generalized Franchetta conjecture for some hyper-Kähler varieties

Abstract

We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten. 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-Macrì-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. 1 2 LIE FU, ROBERT LATERVEER, AND CHARLES VIAL Hyper-Kähler varieties. It was first conjectured by O'Grady [O'G13] that the universal family of K3 surfaces of given genus over the corresponding moduli space satisfies the Franchetta property. By using Mukai models, this was proved for certain families of K3 surfaces of low genus by Pavic-Shen-Yin [PSY17]. By investigating the case of the Beauville-Donagi family [BD85] of Fano varieties of lines on smooth cubic fourfolds, we were led in [FLVS19] to ask whether O'Grady's conjecture holds more generally for hyper-Kähler varieties : Conjecture 1 (Generalized Franchetta conjecture for hyper-Kähler varieties [FLVS19]). Let F be the moduli stack of a locally complete family of polarized hyper-Kähler varieties. Then the universal family X → F satisfies the Franchetta property.It might furthermore be the case that, for some positive integers n, the relative n-powers X n /F → F satisfy the Franchetta property. This was proved for instance in the case n = 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 first main object of study of this paper is about the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten [LLSvS17], subsequently referred to as LLSS eightfolds. An LLSS eightfold is constructed from the space of twisted cubic curves on a smooth cubic fourfold not containing a plane. The following result, which is the first main result of this paper, completes our previous work [FLVS19, Theorem 1.11] where the Franchetta property was established for 0-cycles and codimension-2 cycles on LLSS eightfolds. Theorem 1. The universal family of LLSS hyper-Kähler eightfolds over the moduli space of smooth cubic fourfolds not containing a plane satisfies the Franchetta property.As already observed in [FLVS19, Proposition 2.5], the generalized Franchetta conjecture for a family of hyper-Kähler varieties implies the Beauville-Voisin conjecture [Voi08] for the very general member of the family : Corollary 1. Let Z be an LLSS hyper-Kähler eightfold. Then the Q-subalgebra R * (Z) := h, c j (Z) ⊂ CH * (Z) generated by the natural polarization h and the C...

The Chow Ring of a Cubic Hypersurface

“…Finally, we observe that ∆ 2 X is rationally equivalent to a multiple of π * 1 h n · π * 2 h n , since X is rationally connected. Thus, the generating set (6) truncates to a basis (8) π * 1 h r · π * 2 h s , ∆ X (with 0 ≤ r, s ≤ n) for R * (X × X). This is also a basis for the image of the above cycle class map; hence, the lemma.…”

The Generalized Franchetta Conjecture for Some Hyperkähler Fourfolds

“…The Fourier decomposition. 2 Results of this kind can be found in [12], which was written after the present paper.…”

Algebraic cycles on certain hyperkähler fourfolds with an order 3 non-symplectic automorphism II

“…Since σ respects the bigrading A * ( * ) (X) (proposition 3.2), theorem 3.1 or theorem 4.1 implies that A 2 (X) σ ⊂ A 2 (0) (X) . Since A 2 (0) (X) · A 1 (X) ⊂ A 3 (0) (X) (this is [16,Proposition A.7], which improves upon [41,Proposition 22.7]), it follows that a ∈ A 3 (0) (X) . But A 3 (0) (X) injects into cohomology under the cycle class map [41].…”