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.
Nous démontrons la conjecture de Franchetta généralisée pour la famille localement complète de variétés hyper-kählériennes de dimension 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 . 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.
Accepted:
Published online:
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
Lie Fu 1; Robert Laterveer 2; Charles Vial 3
@article{JEP_2021__8__1065_0, author = {Lie Fu and Robert Laterveer and Charles Vial}, title = {The generalized {Franchetta} conjecture for some {hyper-K\"ahler} varieties, {II}}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1065--1097}, publisher = {\'Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.166}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.166/} }
TY - JOUR AU - Lie Fu AU - Robert Laterveer AU - Charles Vial TI - The generalized Franchetta conjecture for some hyper-Kähler varieties, II JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 1065 EP - 1097 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.166/ DO - 10.5802/jep.166 LA - en ID - JEP_2021__8__1065_0 ER -
%0 Journal Article %A Lie Fu %A Robert Laterveer %A Charles Vial %T The generalized Franchetta conjecture for some hyper-Kähler varieties, II %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 1065-1097 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.166/ %R 10.5802/jep.166 %G en %F JEP_2021__8__1065_0
Lie Fu; Robert Laterveer; Charles Vial. The generalized Franchetta conjecture for some hyper-Kähler varieties, II. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1065-1097. doi : 10.5802/jep.166. https://jep.centre-mersenne.org/articles/10.5802/jep.166/
[AHLH18] - “Existence of moduli spaces for algebraic stacks”, 2018 | arXiv
[AHR20] - “A Luna étale slice theorem for algebraic stacks”, Ann. of Math. (2) 191 (2020) no. 3, p. 675-738 | DOI | Zbl
[And04] - Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas & Synthèses, vol. 17, Société Mathématique de France, Paris, 2004 | Zbl
[Ara06] - “Motivation for Hodge cycles”, Adv. Math. 207 (2006) no. 2, p. 762-781 | DOI | MR | Zbl
[AT14] - “Hodge theory and derived categories of cubic fourfolds”, Duke Math. J. 163 (2014), p. 1885-1927 | DOI | MR | Zbl
[BD85] - “La variété des droites d’une hypersurface cubique de dimension ”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) no. 14, p. 703-706 | Zbl
[BLM + 17] - “Stability conditions on Kuznetsov components”, Ann. Sci. École Norm. Sup. (4) (2017), to appear | arXiv
[BLM + 21] - “Stability conditions in families”, Publ. Math. Inst. Hautes Études Sci. (2021) | arXiv | DOI
[BM14a] - “MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations”, Invent. Math. 198 (2014) no. 3, p. 505-590 | DOI | MR | Zbl
[BM14b] - “Projectivity and birational geometry of Bridgeland moduli spaces”, J. Amer. Math. Soc. 27 (2014) no. 3, p. 707-752 | DOI | MR | Zbl
[Bri07] - “Stability conditions on triangulated categories”, Ann. of Math. (2) 166 (2007) no. 2, p. 317-345 | DOI | MR | Zbl
[Bri08] - “Stability conditions on K3 surfaces”, Duke Math. J. 141 (2008) no. 2, p. 241-291 | DOI | MR | Zbl
[BV04] - “On the Chow ring of a K3 surface”, J. Algebraic Geom. 13 (2004) no. 3, p. 417-426 | DOI | MR | Zbl
[Bül20] - “Motives of moduli spaces on K3 surfaces and of special cubic fourfolds”, Manuscripta Math. 161 (2020) no. 1-2, p. 109-124 | DOI | MR | Zbl
[Dia19] - “The Chow ring of a cubic hypersurface”, Internat. Math. Res. Notices (2019), article ID rnz299 | DOI
[DK19] - “Gushel-Mukai varieties: linear spaces and periods”, Kyoto J. Math. 40 (2019), p. 5-57 | DOI
[DV10] - “Hyper-Kähler fourfolds and Grassmann geometry”, J. reine angew. Math. 649 (2010), p. 63-87 | DOI | Zbl
[FFZ21] - “On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties”, Commun. Contemp. Math. 23 (2021) no. 4, p. 2050034, 50 | DOI | Zbl
[FLV21] - “Multiplicative Chow-Künneth decompositions and varieties of cohomological K3 type”, Ann. Mat. Pura Appl. (4) (2021), to appear
[FLVS19] - “The generalized Franchetta conjecture for some hyper-Kähler varieties”, J. Math. Pures Appl. (9) 130 (2019), p. 1-35 | Zbl
[Ful98] - Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1998 | MR | Zbl
[FV20] - “Distinguished cycles on varieties with motive of Abelian type and the section property”, J. Algebraic Geom. 29 (2020) no. 1, p. 53-107 | DOI | MR | Zbl
[GG03] - “An interesting -cycle”, Duke Math. J. 119 (2003) no. 2, p. 261-313 | DOI | MR | Zbl
[GG12] - “Motives and representability of algebraic cycles on threefolds over a field”, J. Algebraic Geom. 21 (2012) no. 2, p. 347-373 | DOI | MR | Zbl
[GK14] - “Symmetric and exterior powers of categories”, Transform. Groups 19 (2014) no. 1, p. 57-103 | DOI | MR | Zbl
[GS14] - “The Fano variety of lines and rationality problem for a cubic hypersurface”, 2014 | arXiv
[Gus83] - “On Fano varieties of genus ”, Izv. Math. 21 (1983) no. 3, p. 445-459 | DOI | Zbl
[Huy19] - “The geometry of cubic hypersurfaces”, 2019, Notes available from http://www.math.uni-bonn.de/people/huybrech/
[HW89] - “On the decomposition of Brauer’s centralizer algebras”, J. Algebra 121 (1989) no. 2, p. 409-445 | DOI | MR | Zbl
[Jan92] - “Motives, numerical equivalence, and semi-simplicity”, Invent. Math. 107 (1992) no. 3, p. 447-452 | DOI | MR | Zbl
[Kim05] - “Chow groups are finite dimensional, in some sense”, Math. Ann. 331 (2005) no. 1, p. 173-201 | DOI | MR | Zbl
[Kim09] - “Surjectivity of the cycle map for Chow motives”, in Motives and algebraic cycles, Fields Inst. Commun., vol. 56, American Mathematical Society, Providence, RI, 2009, p. 157-165 | MR | Zbl
[KP18] - “Derived categories of Gushel-Mukai varieties”, Compositio Math. 154 (2018), p. 1362-1406 | DOI | MR | Zbl
[Kuz10] - Derived categories of cubic fourfolds, Progress in Math. 282 (2010), p. 219-243 | DOI | MR | Zbl
[Lat17a] - “Algebraic cycles on Fano varieties of some cubics”, Results Math. 72 (2017) no. 1-2, p. 595-616 | DOI | MR | Zbl
[Lat17b] - “A remark on the motive of the Fano variety of lines of a cubic”, Ann. Math. Qué. 41 (2017) no. 1, p. 141-154 | DOI | MR | Zbl
[Lat21] - “On the Chow ring of some Lagrangian fibrations”, Bull. Belg. Math. Soc. Simon Stevin (2021), to appear, arXiv:2105.06857
[Lie06] - “Moduli of complexes on a proper morphism”, J. Algebraic Geom. 15 (2006) no. 1, p. 175-206 | DOI | MR | Zbl
[LLMS18] - “Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects”, J. Math. Pures Appl. (9) 114 (2018), p. 85-117 | DOI | MR | Zbl
[LLSvS17] - “Twisted cubics on cubic fourfolds”, J. reine angew. Math. 731 (2017), p. 87-128 | DOI | MR | Zbl
[LPZ18] - “Twisted cubics on cubic fourfolds and stability conditions”, 2018 | arXiv
[LPZ20] - “Elliptic quintics on cubic fourfolds, O’Grady 10, and Lagrangian fibrations”, 2020 | arXiv
[LS06] - “La singularité de O’Grady”, J. Algebraic Geom. 15 (2006) no. 4, p. 753-770 | DOI | Zbl
[Mar12] - “Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces”, J. reine angew. Math. 544 (2012), p. 61-82 | DOI | MR | Zbl
[MNP13] - Lectures on the theory of pure motives, University Lect. Series, vol. 61, American Mathematical Society, Providence, RI, 2013 | DOI | MR | Zbl
[MS19] - “Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces”, in Birational geometry of hypersurfaces, Lect. Notes of the UMI, vol. 26, Springer, 2019 | DOI | Zbl
[MT15] - “From exceptional collections to motivic decompositions via noncommutative motives”, J. reine angew. Math. 701 (2015), p. 153-167 | DOI | MR | Zbl
[Muk84] - “Symplectic structure of the moduli space of sheaves on an abelian or K3 surface”, Invent. Math. 77 (1984) no. 1, p. 101-116 | DOI | MR | Zbl
[Muk89] - “Biregular classification of Fano -folds and Fano manifolds of coindex ”, Proc. Nat. Acad. Sci. U.S.A. 86 (1989) no. 9, p. 3000-3002 | DOI | MR | Zbl
[MZ16] - “Birational geometry of singular moduli spaces of O’Grady type”, Adv. Math. 296 (2016), p. 210-267 | DOI | MR | Zbl
[MZ20] - “On the group of zero-cycles of holomorphic symplectic varieties”, Épijournal de Géom. Alg. 4 (2020), article ID 3, 5 pages | DOI | MR | Zbl
[Ouc17] - “Lagrangian embeddings of cubic fourfolds containing a plane”, Compositio Math. 153 (2017) no. 5, p. 947-972 | DOI | MR | Zbl
[O’G99] - “Desingularized moduli spaces of sheaves on a K3”, J. reine angew. Math. 512 (1999), p. 49-117 | DOI | MR | Zbl
[O’G03] - “A new six-dimensional irreducible symplectic variety”, J. Algebraic Geom. 12 (2003) no. 3, p. 435-505 | DOI | MR | Zbl
[O’G13] - “Moduli of sheaves and the Chow group of K3 surfaces”, J. Math. Pures Appl. (9) 100 (2013) no. 5, p. 701-718 | DOI | MR | Zbl
[O’S11] - “Algebraic cycles on an abelian variety”, J. reine angew. Math. 654 (2011), p. 1-81 | DOI | MR | Zbl
[Pop18] - “Twisted cubics and quadruples of points on cubic surfaces”, 2018 | arXiv
[PPZ19] - “Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties”, 2019 | arXiv
[PR13] - “Deformation of the O’Grady moduli spaces”, J. reine angew. Math. 678 (2013), p. 1-34 | DOI | MR | Zbl
[PSY17] - “On O’Grady’s generalized Franchetta conjecture”, Internat. Math. Res. Notices (2017) no. 16, p. 4971-4983 | DOI | MR | Zbl
[Rie14] - “On the Chow ring of birational irreducible symplectic varieties”, Manuscripta Math. 145 (2014) no. 3-4, p. 473-501 | DOI | MR | Zbl
[SV16a] - The Fourier transform for certain hyperkähler fourfolds, Mem. Amer. Math. Soc., vol. 240, no. 1139, American Mathematical Society, Providence, RI, 2016 | DOI | Zbl
[SV16b] - “The motive of the Hilbert cube ”, Forum Math. Sigma 4 (2016), article ID e30, 55 pages | DOI | MR | Zbl
[Tav11] - “The tautological ring of ”, Ann. Inst. Fourier (Grenoble) 61 (2011) no. 7, p. 2751-2779 | DOI | Numdam | MR | Zbl
[Tav14] - “The tautological ring of the moduli space ”, Internat. Math. Res. Notices (2014) no. 24, p. 6661-6683 | DOI | MR | Zbl
[Tav18] - “Tautological classes on the moduli space of hyperelliptic curves with rational tails”, J. Pure Appl. Algebra 222 (2018) no. 8, p. 2040-2062 | DOI | MR | Zbl
[Via10] - “Pure motives with representable Chow groups”, Comptes Rendus Mathématique 348 (2010) no. 21-22, p. 1191-1195 | DOI | MR | Zbl
[Via13] - “Projectors on the intermediate algebraic Jacobians”, New York J. Math. 19 (2013), p. 793-822 | MR | Zbl
[Voi08] - “On the Chow ring of certain algebraic hyper-Kähler manifolds”, Pure Appl. Math. Q 4 (2008) no. 3, p. 613-649, Special Issue: In honor of Fedor Bogomolov. Part 2 | DOI | Zbl
[Voi12] - “Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces”, Geom. Topol. 16 (2012) no. 1, p. 433-473 | DOI | MR | Zbl
[Voi16] - “Remarks and questions on coisotropic subvarieties and -cycles of hyper-Kähler varieties”, in K3 surfaces and their moduli, Progress in Math., vol. 315, Birkhäuser/Springer, 2016, p. 365-399 | DOI | Zbl
[Voi17] - “On the universal group of cubic hypersurfaces”, J. Eur. Math. Soc. (JEMS) 19 (2017) no. 6, p. 1619-1653 | DOI | Zbl
[Voi18] - “Hyper-Kähler compactification of the intermediate Jacobian fibration of a cubic fourfold: the twisted case”, in Local and global methods in algebraic geometry, Contemp. Math., vol. 712, American Mathematical Society, Providence, RI, 2018, p. 341-355 | DOI | Zbl
[Yin15a] - “Finite-dimensionality and cycles on powers of K3 surfaces”, Comment. Math. Helv. 90 (2015) no. 2, p. 503-511 | DOI | MR | Zbl
[Yin15b] - “The generic nontriviality of the Faber-Pandharipande cycle”, Internat. Math. Res. Notices (2015) no. 5, p. 1263-1277 | DOI | MR | Zbl
[Yos01] - “Moduli spaces of stable sheaves on abelian surfaces”, Math. Ann. 321 (2001) no. 4, p. 817-884 | DOI | MR | Zbl
[YY14] - “Bridgeland’s stabilities on abelian surfaces”, Math. Z. 276 (2014) no. 1-2, p. 571-610 | DOI | MR | Zbl
Cited by Sources: