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 hyperkahler 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.
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