Hilbert squares of K3 surfaces and Debarre-Voisin varieties
[Schémas de Hilbert ponctuels de surfaces K3 et variétés de Debarre-Voisin]
Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 653-710.

Les variétés hyper-kählériennes de Debarre-Voisin sont construites à l’aide de 3-formes alternées sur un espace vectoriel complexe de dimension 10, que nous appelons des trivecteurs. Elles présentent de nombreuses analogies avec les variétés de Beauville-Donagi qui sont construites en partant d’une cubique de dimension 4. Nous étudions dans cet article différents trivecteurs dont la variété de Debarre-Voisin associée est dégénérée au sens où elle est soit réductible, soit de dimension excessive. Nous montrons que, sous une spécialisation d’un trivecteur général en de tels trivecteurs, les variétés de Debarre-Voisin correspondantes se spécialisent en des variétés hyper-kählériennes lisses, birationnellement isomorphes au schéma de Hilbert des paires de points sur une surface K3.

Debarre-Voisin hyperkähler fourfolds are built from alternating 3-forms on a 10-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general 1-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.

Reçu le : 2019-01-15
Accepté le : 2020-01-28
Publié le : 2020-04-02
DOI : https://doi.org/10.5802/jep.125
Classification : 14J32,  14J35,  14M15,  14J70,  14J28
Mots clés: Variétés hyper-kählériennes, trivecteurs, espaces de modules, schémas de Hilbert ponctuels de surfaces K3
@article{JEP_2020__7__653_0,
     author = {Olivier Debarre and Fr\'ed\'eric Han and Kieran O'Grady and Claire Voisin},
     title = {Hilbert squares of K3 surfaces and Debarre-Voisin varieties},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {653-710},
     doi = {10.5802/jep.125},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__653_0/}
}
Olivier Debarre; Frédéric Han; Kieran O’Grady; Claire Voisin. Hilbert squares of K3 surfaces and Debarre-Voisin varieties. Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 653-710. doi : 10.5802/jep.125. https://jep.centre-mersenne.org/item/JEP_2020__7__653_0/

[Apo14] A. Apostolov - “Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected”, Ann. Inst. Fourier (Grenoble) 64 (2014) no. 1, p. 189-202 | Article | Numdam | MR 3330546 | Zbl 1329.14075

[BD85] A. Beauville & R. Donagi - “La variété des droites d’une hypersurface cubique de dimension 4”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) no. 14, p. 703-706

[BM14] A. Bayer & E. Macrì - “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 | Article | MR 3279532 | Zbl 1308.14011

[Bou90] N. Bourbaki - Groupes et algèbres de Lie, chapitres 7 et 8. Éléments de mathématiques, Masson, Paris, 1990 | Zbl 1181.17001

[CS16] I. Cheltsov & C. Shramov - Cremona groups and the icosahedron, Monographs and Research Notes in Math., CRC Press, Boca Raton, FL, 2016 | Zbl 1328.14003

[Dan01] G. Danila - “Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface”, J. Algebraic Geom. 10 (2001) no. 2, p. 247-280 | MR 1811556 | Zbl 0988.14011

[Dan07] G. Danila - “Sections de la puissance tensorielle du fibré tautologique sur le schéma de Hilbert des points d’une surface”, Bull. London Math. Soc. 39 (2007) no. 2, p. 311-316 | Article | MR 2323464 | Zbl 1122.14002

[Deb18] O. Debarre - “Hyperkähler manifolds”, 2018 | arXiv:1810.02087

[DIM15] O. Debarre, A. Iliev & L. Manivel - “Special prime Fano fourfolds of degree 10 and index 2”, in Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, p. 123-155 | Article | MR 3380447 | Zbl 1326.14094

[DM19] O. Debarre & E. Macrì - “On the period map for polarized hyperkähler fourfolds”, Internat. Math. Res. Notices (2019) no. 22, p. 6887-6923 | Article | Zbl 07144226

[Dol12] I. V. Dolgachev - Classical algebraic geometry. A modern view, Cambridge University Press, Cambridge, 2012 | Article | MR 2964027 | Zbl 1252.14001

[DV10] O. Debarre & C. Voisin - “Hyper-Kähler fourfolds and Grassmann geometry”, J. reine angew. Math. 649 (2010), p. 63-87 | Article | Zbl 1217.14028

[GHS07] V. A. Gritsenko, K. Hulek & G. K. Sankaran - “The Kodaira dimension of the moduli of K3 surfaces”, Invent. Math. 169 (2007) no. 3, p. 519-567 | Article | MR 2336040 | Zbl 1128.14027

[GHS10] V. Gritsenko, K. Hulek & G. K. Sankaran - “Moduli spaces of irreducible symplectic manifolds”, Compositio Math. 146 (2010) no. 2, p. 404-434 | Article | MR 2601632 | Zbl 1230.14051

[GHS13] V. Gritsenko, K. Hulek & G. K. Sankaran - “Moduli of K3 surfaces and irreducible symplectic manifolds”, in Handbook of moduli. Vol. I, Adv. Lect. Math., vol. 24, Int. Press, Somerville, MA, 2013, p. 459-526 | MR 3184170 | Zbl 1322.14004

[GS] D. Grayson & M. Stillman - “Macaulay2, a software system for research in algebraic geometry”, available at https://faculty.math.illinois.edu/Macaulay2/

[Han] F. Han - “Computations with Macaulay2”, http://webusers.imj-prg.fr/~frederic.han/recherche/documents/debarre-ogrady-han-voisin.m2

[Has00] B. Hassett - “Special cubic fourfolds”, Compositio Math. 120 (2000) no. 1, p. 1-23 | Article | MR 1738215 | Zbl 0956.14031

[Hiv11] P. Hivert - “Equations of some wonderful compactifications”, Ann. Inst. Fourier (Grenoble) 61 (2011) no. 5, p. 2121-2138 (2012) | Article | Numdam | MR 2961850 | Zbl 1298.14050

[Hor77] E. Horikawa - “Surjectivity of the period map of K3 surfaces of degree 2”, Math. Ann. 228 (1977) no. 2, p. 113-146 | Article

[HT09] B. Hassett & Y. Tschinkel - “Moving and ample cones of holomorphic symplectic fourfolds”, Geom. Funct. Anal. 19 (2009) no. 4, p. 1065-1080 | Article | MR 2570315 | Zbl 1183.14058

[Huy12] D. Huybrechts - “A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky]”, in Séminaire Bourbaki, Astérisque, vol. 348, Société Mathématique de France, Paris, 2012, p. 375-403, Exp. no. 1040 | Zbl 1272.32014

[Isk77] V. A. Iskovskih - “Fano threefolds. I”, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) no. 3, p. 516-562, English transl.: Math. USSR Izv. 11 (1977), p. 485–527 | MR 463151

[KLSV18] J. Kollár, R. Laza, G. Saccà & C. Voisin - “Remarks on degenerations of hyper-Kähler manifolds”, Ann. Inst. Fourier (Grenoble) 68 (2018) no. 7, p. 2837-2882 | Article | Zbl 07058392

[Kru14] A. Krug - “Tensor products of tautological bundles under the Bridgeland-King-Reid-Haiman equivalence”, Geom. Dedicata 172 (2014), p. 245-291 | Article | MR 3253782 | Zbl 1337.14005

[Laz09] R. Laza - “The moduli space of cubic fourfolds”, J. Algebraic Geom. 18 (2009), p. 511-545 | Article | MR 2496456 | Zbl 1169.14026

[Laz10] R. Laza - “The moduli space of cubic fourfolds via the period map”, Ann. of Math. (2) 172 (2010) no. 1, p. 673-711 | Article | MR 2680429 | Zbl 1201.14026

[Loo03] E. Looijenga - “Compactifications defined by arrangements. II. Locally symmetric varieties of type IV”, Duke Math. J. 119 (2003) no. 3, p. 527-588 | Article | MR 2003125 | Zbl 1079.14045

[Loo09] E. Looijenga - “The period map for cubic fourfolds”, Invent. Math. 177 (2009) no. 1, p. 213-233 | Article | MR 2507640 | Zbl 1177.32010

[Lun75] D. Luna - “Adhérences d’orbite et invariants”, Invent. Math. 29 (1975) no. 3, p. 231-238 | Article | MR 376704 | Zbl 0315.14018

[Mar11] E. Markman - “A survey of Torelli and monodromy results for holomorphic-symplectic varieties”, in Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, p. 257-322 | Article | MR 2964480 | Zbl 1229.14009

[Muk84] S. Mukai - “Symplectic structure of the moduli space of sheaves on an abelian or K3 surface”, Invent. Math. 77 (1984) no. 1, p. 101-116 | Article | MR 751133 | Zbl 0565.14002

[Muk88] S. Mukai - “Curves, K3 surfaces and Fano 3-folds of genus 10”, in Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, p. 357-377

[Muk06] S. Mukai - “Polarized K3 surfaces of genus thirteen”, in Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, p. 315-326 | Article | MR 2310254 | Zbl 1117.14040

[Muk16] S. Mukai - “K3 surfaces of genus sixteen”, in Minimal models and extremal rays (Kyoto, 2011), Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, Tokyo, 2016, p. 379-396 | Article | MR 3618267 | Zbl 1369.14049

[Nag64] T. Nagell - Introduction to number theory, Second edition, Chelsea Publishing Co., New York, 1964

[O’G15] K. G. O’Grady - “Periods of double EPW-sextics”, Math. Z. 280 (2015) no. 1-2, p. 485-524 | Article | MR 3343917 | Zbl 1327.14061

[O’G16] K. G. O’Grady - Moduli of double EPW-sextics, Mem. Amer. Math. Soc., vol. 240, no. 1136, American Mathematical Society, Providence, RI, 2016 | Article | MR 3460099 | Zbl 1408.14116

[O’G19] K. G. O’Grady - “Modular sheaves on hyperkähler varieties”, 2019 | arXiv:1912.02659

[Sha80] J. Shah - “A complete moduli space for K3 surfaces of degree 2”, Ann. of Math. (2) 112 (1980) no. 3, p. 485-510 | Article

[Spr09] T. A. Springer - Linear algebraic groups, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009 | Zbl 1202.20048

[vdD12] B. van den Dries - Degenerations of cubic fourfolds and holomorphic symplectic geometry, Ph. D. Thesis, Universiteit Utrecht, 2012, available at https://dspace.library.uu.nl/handle/1874/233790

[Ver13] M. Verbitsky - “Mapping class group and a global Torelli theorem for hyperkähler manifolds”, Duke Math. J. 162 (2013) no. 15, p. 2929-2986, Appendix A by Eyal Markman | Article | Zbl 1295.53042

[Vie90] E. Viehweg - “Weak positivity and the stability of certain Hilbert points. III”, Invent. Math. 101 (1990) no. 3, p. 521-543 | Article | MR 1062794 | Zbl 0746.14014

[Wan14] M. Wandel - “Stability of tautological bundles on the Hilbert scheme of two points on a surface”, Nagoya Math. J. 214 (2014), p. 79-94 | Article | MR 3211819 | Zbl 1319.14049