Hostname: page-component-5b777bbd6c-6lqsf Total loading time: 0 Render date: 2025-06-19T19:41:40.526Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards a modular construction of OG10

Part of: Symplectic geometry, contact geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  08 November 2024

Alessio Bottini
Alessio Bottini*
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy bottini@math.uni-bonn.de Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, Rue Michel Magat, Bât. 307, 91405 Orsay, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$-type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

Keywords

Hyper-Kähler manifoldsstable vector bundlesmoduli spaces

MSC classification

Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 53D30: Symplectic structures of moduli spaces
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 10 , October 2024 , pp. 2496 - 2529
Check for updates
Copyright
© The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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.)

Article purchase

Temporarily unavailable

Footnotes

Current address: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

References

Addington, N., Donovan, W. and Meachan, C., Moduli spaces of torsion sheaves on $K3$ surfaces and derived equivalences, J. Lond. Math. Soc. (2) 93 (2016), 846–865.CrossRefGoogle Scholar
Arinkin, D., Autoduality of compactified Jacobians for curves with plane singularities, J. Algebraic Geom. 22 (2013), 363–388.CrossRefGoogle Scholar
Beckmann, T., Atomic objects on hyper-Kähler manifolds, Preprint (2022), arXiv:2202.01184.Google Scholar
Beckmann, T., Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure, Compos. Math. 159 (2023), 109–152.CrossRefGoogle Scholar
Beckmann, T. and Song, J., Second Chern class and Fujiki constants of hyper-Kähler manifolds, Preprint (2022), arXiv:2201.07767.Google Scholar
Buchweitz, R. and Flenner, H., A semiregularity map for modules and applications to deformations, Compos. Math. 137 (2003), 135–210.CrossRefGoogle Scholar
Căldăraru, A., Katz, S. and Sharpe, E., D-branes, $B$ fields, and Ext groups, Adv. Theor. Math. Phys. 7 (2003), 381–404.CrossRefGoogle Scholar
Collino, A., The fundamental group of the Fano surface, in Algebraic threefolds (Varenna, 1981), Lecture Notes in Mathematics (Springer, Berlin–New York, 1982), 209–220.CrossRefGoogle Scholar
Hartshorne, R., Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.CrossRefGoogle Scholar
Huybrechts, D., Birational symplectic manifolds and their deformations, J. Differential Geom. 45 (1997), 488–513.CrossRefGoogle Scholar
Huybrechts, D., Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs (Clarendon Press, Oxford, 2006).CrossRefGoogle Scholar
Huybrechts, D. and Thomas, R., $\mathbb {P}$-objects and autoequivalences of derived categories, Math. Res. Lett. 13 (2006), 87–98.CrossRefGoogle Scholar
Kobayashi, S., Simple vector bundles over symplectic Kähler manifolds, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 21–24.CrossRefGoogle Scholar
Kuznetsov, A. and Markushevich, D., Symplectic structures on moduli spaces of sheaves via the Atiyah class, J. Geom. Phys. 59 (2009), 843–860.CrossRefGoogle Scholar
Looijenga, E. and Lunts, V., A Lie algebra attached to a projective variety, Invent. Math. 129 (1997), 361–412.CrossRefGoogle Scholar
Mladenov, B., Degeneration of spectral sequences and complex Lagrangian submanifolds, Preprint (2019), arXiv:1907.04742.Google Scholar
Markman, E., The Beauville–Bogomolov class as a characteristic class, J. Algebraic Geom. 29 (2020), 199–245.CrossRefGoogle Scholar
Markman, E., Stable vector bundles on a hyper-Kähler manifold with a rank 1 obstruction map are modular, Kyoto J. Math., to appear. Preprint (2023), arXiv:2107.13991.Google Scholar
Meazzini, F. and Onorati, C., Hyper-holomorphic connections on vector bundles on hyper-Kähler manifolds, Math. Z. 303 (2023), Paper No. 17.CrossRefGoogle Scholar
Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), 101–116.CrossRefGoogle Scholar
O'Grady, K., The weight-two Hodge structure of moduli spaces of sheaves on a $K3$ surface, J. Algebraic Geom. 6 (1997), 599–644.Google Scholar
O'Grady, K., Desingularized moduli spaces of sheaves on a $K3$, J. Reine Angew. Math. 512 (1999), 49–117.CrossRefGoogle Scholar
O'Grady, K., A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), 435–505.CrossRefGoogle Scholar
O'Grady, K., Moduli of sheaves on $K3$'s and higher dimensional HK varieties, Preprint (2021), arXiv:2109.07425.Google Scholar
O'Grady, K., Modular sheaves on hyperkähler varieties, Algebr. Geom. 9 (2022), 1–38.Google Scholar
Serre, J.-P., Algèbre locale. Multiplicités, Lecture Notes in Mathematics (Springer, Berlin– New York, 1965).Google Scholar
Taelman, L., Derived equivalences of hyperkähler varieties, Geom. Topol. 27 (2023), 2649–2693.CrossRefGoogle Scholar
Toda, Y., Deformations and Fourier-Mukai transforms, J. Differential Geom. 81 (2009), 197–224.CrossRefGoogle Scholar
van den Dries, B., Degenerations of cubic fourfolds and holomorphic symplectic geometry, PhD Thesis, Universiteit Utrecht (2012).Google Scholar
Verbitsky, M., Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), 601–611.CrossRefGoogle Scholar
Verbitsky, M., Hyperholomorphic sheaves and new examples of hyperkaehler manifolds, in Hyperkähler manifolds, Mathematical Physics (Somerville), vol. 12 (International Press, Somerville, MA, 1999).Google Scholar
Verbitsky, M., Coherent sheaves on general $K3$ surfaces and tori, Pure Appl. Math. Q. 4 (2008), 651–714.CrossRefGoogle Scholar
Verbitsky, M., Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013), 2929–2986.CrossRefGoogle Scholar
Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817–884.CrossRefGoogle Scholar