Hyperkahler manifolds from the Tits-Freudenthal square

TL;DR

This paper introduces a novel geometric construction linking Lie algebras of types G2, D4, F4, E6, E7, E8 to families of hyperkähler fourfolds via cycles on hyperplane sections of specific varieties, inspired by Freudenthal's Legendrian varieties.

Contribution

It establishes a new method to associate hyperkähler manifolds to Lie algebras using geometric cycles modeled on Legendrian varieties within the Tits-Freudenthal square.

Findings

Constructs families of hyperkähler fourfolds from Lie algebras.

Connects Lie algebra types to geometric cycle families.

Provides a geometric interpretation of the Tits-Freudenthal square.

Abstract

We suggest a way to associate to each Lie algebra of type G2, D4, F4, E6, E7, E8 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.