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Hyperkähler manifolds from the Tits–Freudenthal magic square

  • Atanas Iliev
  • Laurent Manivel
Research Article
  • 9 Downloads

Abstract

We suggest a way to associate to each Lie algebra of type \(G_2,D_4,F_4,E_6\), \(E_7,E_8\) a family of polarized hyperkähler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasi-homogeneous varieties. These cycles are modeled on the Legendrian varieties studied by Freudenthal in his geometric approach to the celebrated Tits–Freudenthal magic square of Lie algebras.

Keywords

Fano varieties Hyperkähler varieties Legendrian varieties Tits–Freudenthal square 

Mathematics Subject Classification

14J28 14J45 14M15 

Notes

Acknowledgements

We thank Alexander Kuznetsov, Dmitri Orlov, Kieran O’Grady, Grzegorz Kapustka and Michał Kapustka for their useful hints and comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsSeoul National UniversitySeoulKorea
  2. 2.Institut de Mathématiques de Toulouse, UMR 5219Université de Toulouse, CNRS, UPS IMTToulouse Cedex 9France

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