The generalized Franchetta conjecture for some hyper-Kähler varieties, II

Fu, Lie; Laterveer, Robert; Vial, Charles

doi:10.5802/jep.166

The generalized Franchetta conjecture for some hyper-Kähler varieties, II
[La conjecture de Franchetta généralisée pour certaines variétés hyper-kählériennes, II]

Fu, Lie ¹ ; Laterveer, Robert ² ; Vial, Charles ³

¹ Institut Camille Jordan, Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
² CNRS - IRMA, Université de Strasbourg 7 rue René-Descartes, 67084 Strasbourg Cedex, France
³ Fakultät für Mathematik, Universität Bielefeld Postfach 100131, 33501 Bielefeld, Germany

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1065-1097.

Résumé
Abstract

Nous démontrons la conjecture de Franchetta généralisée pour la famille localement complète de variétés hyper-kählériennes de dimension $8$ construite par Lehn-Lehn-Sorger-van Straten (LLSS). Comme corollaire, nous établissons la conjecture de Beauville-Voisin pour les variétés LLSS très générales. Notre stratégie consiste à utiliser la description récente de ces variétés LLSS comme espaces de modules d’objets semistables (au sens de Bridgeland) dans la composante de Kuznetsov de la catégorie dérivée d’hypersurfaces cubiques, et notamment la version relative due à Bayer-Lahoz-Macrì-Nuer-Perry-Stellari, pour nous réduire à la propriété de Franchetta pour les puissances relatives quatrièmes d’hypersurfaces cubiques de dimension $4$ . Nos résultats nous permettent également de décrire le motif de Chow de la variété de Fano des droites sur une hypersurface cubique lisse en termes du motif de Chow de l’hypersurface cubique.

We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler 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 Bridgeland 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.

Reçu le : 2020-09-21
Accepté le : 2021-05-19
Publié le : 2021-05-25

DOI : 10.5802/jep.166

Classification : 14C25, 14C15, 14J42, 14J28, 14F08, 14J70, 14D20, 14H10
Keywords: Chow ring, motives, hyper-Kähler varieties, cubic hypersurfaces, Franchetta conjecture, Beauville–Voisin conjecture, derived categories, stability conditions, Kuznetsov component, moduli space of sheaves, tautological ring
Mot clés : Anneau de Chow, motifs, variétés hyper-kählériennes, hypersurfaces cubiques, conjecture de Franchetta, conjecture de Beauville-Voisin, catégories dérivées, conditions de stabilité, composante de Kuznetsov, espaces de modules de faisceaux, anneau tautologique

Affiliations des auteurs :

Fu, Lie ¹ ; Laterveer, Robert ² ; Vial, Charles ³

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     title = {The generalized {Franchetta} conjecture for some {hyper-K\"ahler} varieties, {II}},
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Fu, Lie; Laterveer, Robert; Vial, Charles. The generalized Franchetta conjecture for some hyper-Kähler varieties, II. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1065-1097. doi : 10.5802/jep.166. http://www.numdam.org/articles/10.5802/jep.166/

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