Hyperkähler manifolds from the Tits -- Freudenthal magic square
type: Item
instance of: scholarly article
title: Hyperkähler manifolds from the Tits -- Freudenthal magic square
zbMATH Open document ID: 1453.14093
author: Laurent Manivel
author: Atanas Iliev
published in: European Journal of Mathematics
publication date: 2019-11-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1612.08173
review text: 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.
reviewed by: Dino Festi
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 7137465
zbMATH Keywords: Fano varieties
zbMATH Keywords: hyperkähler varieties
zbMATH Keywords: Legendrian varieties
zbMATH Keywords: Tits-Freudenthal square
MaRDI profile type: Publication
OpenAlex ID: W2587404957
arXiv ID: 1612.08173
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DOI: 10.1007/S40879-018-00313-4
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description: scientific article