Hyperkähler manifolds from the Tits -- Freudenthal magic square
Finding examples of compact hyperkähler manifolds is a difficult problem in algebraic geometry. In this paper, the authors propose a new conjectural method 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. It is known that to each simple complex Lie algebra one can associate a projective homogeneous variety \(X_{\textrm{ad}}\), called the adjoint variety; this adjoint variety is covered by a family of special subvarieties, called Legendrian cycles. The authors conjecture that for the considered Lie algebras, it is possible to construct a projective variety \(X\) such that \begin{itemize} \item \(\dim X = \dim X_{\textrm{ad}}\), \item \(X\) is also covered by Legendrian cycles, \item there is a suitable cycle space \(P\) for these Legendrian cycles, \item for a generic hyperplane section \(X_H\) of \(X\), the subspace \(P_H\subset P\) parametrizing the Legendrian cycles contained in \(X_H\) is birational to a hyperkähaler fourfold. \end{itemize} The conjecture is proven and \(P_H\) is explicitly described for the Lie algebras \(G_2, D_4, F_4, E_6\); the cases \(E_7, E_8\) stay open.
scientific article