Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-05T19:03:43.492Z Has data issue: false hasContentIssue false
  • English
  • Français

NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF $K3^{[n]}$-TYPE

Part of: Cycles and subschemes Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  27 February 2020

CHIARA CAMERE Open the ORCID record for CHIARA CAMERE [Opens in a new window] ,
ALBERTO CATTANEO Open the ORCID record for ALBERTO CATTANEO [Opens in a new window]  and
ANDREA CATTANEO Open the ORCID record for ANDREA CATTANEO [Opens in a new window]
CHIARA CAMERE
Affiliation:
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133Milano, Italy email chiara.camere@unimi.it
ALBERTO CATTANEO
Affiliation:
Mathematisches Institut and Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115Bonn, Germany email cattaneo@math.uni-bonn.de
ANDREA CATTANEO
Affiliation:
Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni, 67/a, 50134Firenze, Italy email cattaneo@math.univ-lyon1.fr, andrea.cattaneo@unifi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

MSC classification

Primary: 14J50: Automorphisms of surfaces and higher-dimensional varieties 14C05: Parametrization (Chow and Hilbert schemes) 14C34: Torelli problem
Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 278 - 302
Check for updates
Copyright
© 2020 Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artebani, M., Sarti, A. and Taki, S., K3 surfaces with non-symplectic automorphisms of prime order, Math. Z. 268(1–2) (2011), 507–533; With an appendix by Shigeyuki Kondō. MR 2805445.CrossRefGoogle Scholar
Bayer, A. and Macrì, E., MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math. 198(3) (2014), 505–590. MR 3279532.CrossRefGoogle Scholar
Beauville, A., “Some remarks on Kähler manifolds with c 1 = 0”, in Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progress in Mathematics 39, Birkhäuser Boston, Boston, MA, 1983, 1–26. MR 728605.Google Scholar
Beauville, A., Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18(4) (1983), 755–782; 1984. MR 730926.CrossRefGoogle Scholar
Beauville, A., Antisymplectic involutions of holomorphic symplectic manifolds, J. Topol. 4(2) (2011), 300–304. MR 2805992.CrossRefGoogle Scholar
Boissière, S., Camere, C. and Sarti, A., Classification of automorphisms on a deformation family of hyper-Kähler four-folds by p-elementary lattices, Kyoto J. Math. 56(3) (2016), 465–499. MR 3542771.CrossRefGoogle Scholar
Boissière, S., Cattaneo, An., Markushevich, D. and Sarti, A., On the nonsymplectic involutions of the Hilbert square of a K3 surface, Izv. Math. 83(4) (2019), 731–742.CrossRefGoogle Scholar
Boissière, S., Cattaneo, An., Nieper-Wisskirchen, M. and Sarti, A., “The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface”, in K3 Surfaces and Their Moduli, Progress in Mathematics 315, Birkhäuser/Springer, [Cham], 2016, 1–15. MR 3524162.Google Scholar
Boissière, S., Nieper-Wißkirchen, M. and Sarti, A., Smith theory and irreducible holomorphic symplectic manifolds, J. Topol. 6(2) (2013), 361–390. MR 3065180.CrossRefGoogle Scholar
Boissière, S. and Sarti, A., A note on automorphisms and birational transformations of holomorphic symplectic manifolds, Proc. Amer. Math. Soc. 140(12) (2012), 4053–4062. MR 2957195.10.1090/S0002-9939-2012-11277-8CrossRefGoogle Scholar
Camere, C. and Cattaneo, Al., Non-symplectic automorphisms of odd prime order on manifolds of K3[n] -type, Manuscripta Math. (2019), doi:10.1007/s00229-019-01163-4.CrossRefGoogle Scholar
Camere, C., Kapustka, G., Kapustka, M. and Mongardi, G., Verra fourfolds, twisted sheaves and the last involution, Int. Math. Res. Not. IMRN 2019(21) 6661–6710.CrossRefGoogle Scholar
Cattaneo, Al., Automorphisms of Hilbert schemes of points on a generic projective K3 surface, Math. Nachr. 292(10) (2019), 2137–2152.CrossRefGoogle Scholar
Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 290, Springer, New York, 1999, With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447.CrossRefGoogle Scholar
Dolgachev, I., “Integral quadratic forms: applications to algebraic geometry (after V. Nikulin)”, in Bourbaki Seminar, Vol. 1982/83, Astérisque 105, Soc. Math. France, Paris, 1983, 251–278. MR 728992.Google Scholar
Gritsenko, V., Hulek, K. and Sankaran, G. K., The Kodaira dimension of the moduli of K3 surfaces, Invent. Math. 169(3) (2007), 519–567. MR 2336040.CrossRefGoogle Scholar
Gritsenko, V., Hulek, K. and Sankaran, G. K., Moduli spaces of irreducible symplectic manifolds, Compos. Math. 146(2) (2010), 404–434. MR 2601632.CrossRefGoogle Scholar
Gritsenko, V., Hulek, K. and Sankaran, G. K., “Moduli of K3 surfaces and irreducible symplectic manifolds”, in Handbook of Moduli, Vol. I, Advanced Lectures in Mathematics (ALM) 24, International Press, Somerville, MA, 2013, 459–526. MR 3184170.Google Scholar
Huybrechts, D. and Stellari, P., Equivalences of twisted K3 surfaces, Math. Ann. 332(4) (2005), 901–936. MR 2179782.CrossRefGoogle Scholar
Iliev, A., Kapustka, G., Kapustka, M. and Ranestad, K., Hyper-Kähler fourfolds and Kummer surfaces, Proc. Lond. Math. Soc. (3) 115(6) (2017), 1276–1316. MR 3741852.CrossRefGoogle Scholar
Iliev, A., Kapustka, G., Kapustka, M. and Ranestad, K., EPW cubes, J. Reine Angew. Math. 748 (2019), 241–268. MR 3918436.CrossRefGoogle Scholar
Joumaah, M., Automorphisms of irreducible symplectic manifolds, Ph.D thesis, Leibniz Universität Hannover, 2015.Google Scholar
Lahoz, M., Lehn, M., Macrì, E. and Stellari, P., Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects, J. Math. Pures Appl. (9) 114 (2018), 85–117. MR 3801751.CrossRefGoogle Scholar
Macrì, E. and Stellari, P., Fano varieties of cubic fourfolds containing a plane, Math. Ann. 354(3) (2012), 1147–1176. MR 2983083.CrossRefGoogle Scholar
Markman, E., “A survey of Torelli and monodromy results for holomorphic-symplectic varieties”, in Complex and Differential Geometry, Springer Proceedings in Mathematics 8, Springer, Heidelberg, 2011, 257–322. MR 2964480.CrossRefGoogle Scholar
Mongardi, G., On natural deformations of symplectic automorphisms of manifolds of K3[n] type, C. R. Math. Acad. Sci. Paris 351(13–14) (2013), 561–564. MR 3095106.CrossRefGoogle Scholar
Mongardi, G., A note on the Kähler and Mori cones of hyperkähler manifolds, Asian J. Math. 19(4) (2015), 583–591. MR 3423735.CrossRefGoogle Scholar
Mongardi, G., On the monodromy of irreducible symplectic manifolds, Algebr. Geom. 3(3) (2016), 385–391. MR 3504537.CrossRefGoogle Scholar
Mongardi, G., Towards a classification of symplectic automorphisms on manifolds of K3[n] type, Math. Z. 282(3–4) (2016), 651–662. MR 3473636.10.1007/s00209-015-1557-xCrossRefGoogle Scholar
Mongardi, G., Tari, K. and Wandel, M., Prime order automorphisms of generalised Kummer fourfolds, Manuscripta Math. 155(3–4) (2018), 449–469. MR 3763414.CrossRefGoogle Scholar
Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43(1) (1979), 111–177; 238. MR 525944.Google Scholar
Nikulin, V. V., Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, J. Soviet Math. 22 (1983), 1401–1475.CrossRefGoogle Scholar
Nikulin, V. V., Involutions of integer quadratic forms and their applications to real algebraic geometry, Izv. Akad. Nauk SSSR Ser. Mat. 47(1) (1983), 109–188. MR 688920.Google Scholar
O’Grady, K. G., Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49–117. MR 1703077.CrossRefGoogle Scholar
O’Grady, K. G., Irreducible symplectic 4-folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134(1) (2006), 99–137. MR 2239344.CrossRefGoogle Scholar
Perego, A. and Rapagnetta, A., Deformation of the O’Grady moduli spaces, J. Reine Angew. Math. 678 (2013), 1–34. MR 3056101.10.1515/CRELLE.2011.191CrossRefGoogle Scholar
Rapagnetta, A., On the Beauville form of the known irreducible symplectic varieties, Math. Ann. 340(1) (2008), 77–95. MR 2349768.CrossRefGoogle Scholar
Saint-Donat, B., Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639. MR 0364263.CrossRefGoogle Scholar
van Geemen, B., Some remarks on Brauer groups of K3 surfaces, Adv. Math. 197(1) (2005), 222–247. MR 2166182.CrossRefGoogle Scholar
Verbitsky, M., Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162(15) (2013), 2929–2986; Appendix A by Eyal Markman. MR 3161308.CrossRefGoogle Scholar
Yoshioka, K., “Moduli spaces of twisted sheaves on a projective variety”, in Moduli Spaces and Arithmetic geometry, Advanced Studies in Pure Mathematic 45, Mathematical Soceity Japan, Tokyo, 2006, 1–30. MR 2306170.Google Scholar