Jordan algebraic and differential geometric study of homogeneous complex manifolds

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For any complex simple Lie algebra, the Freudenthal variety is defined as the linear section of the corresponding adjoint variety by the 1-st graded subspace with respect to the complex contact type gradation of the Lie algebra. In this research [KY], the homogeneity of the Freudenthal varieties is proved in a priori way. And a unified description is given for the orbit structure of the first graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra. Then several properties of the Freidenthal varieties as complex projective varieties are derived only from the axioms of symplectic triple systems which are enjoyed by the 1-st graded subspace. Moreover, it provides a short cutting alternating proof for the 1-st graded subspace to enjoy the axioms of symplectic triple systems. These results are based on the construction of all complex simple Lie algebra of rank non less than two from simple complex symplectic triple systems by K.Yamaguti-H.Asano (1975), and vice varsa by H.Asano.(1975). Note that these results avoid case-by-case arguments according to the classification of all complex simple Lie algebras, and that these results are drived by considering substructure of symplectic triple systems. Algebraically, a structure theory of symplectic triple systems is studied by H.Asano (unpublished). These results give a geometric light on this algebraic structure theory of symplectic triple systems.In this research [Y], typical examples are found on two non-isomorphic reductive Lie groups with one dimensional center such that their Lie algebras are isomorphic. In the literature, no proof was published on these examples. This result is firstly observed by considering the result of the orbit decomposition of the 1-st graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra of complex simple Lie algebras of type B and D.