Research on homogeneous projective varieties by Lie algebra and algebraic geometry

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H. Maeda gave the following results :(1) Let E be an ample vector bundle of rank n-2 on a complex projective manifold of dimension n having a section whose zero locus Z is an algebraic surface of Kodaira dimension 1. Then the structure of E is completely determined. This generalizes Sommese and Shepherd-Barron's results on ample divisors.(2) A classification of the polarized varieties (X, E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle E of rank n-1 on X such that E has a section whose zero locus is a smooth elliptic curve. And the property of E is investigated when E is very ample having a section whose zero locus equals a hyperelliptic curve of genus non less than two.(3) In particular, a classification of such (X, E)'s is given when the genus of Z equals two. A classification of the polarized varieties (X, E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle E of rank n-r on X such that E has a section whose zero locus Z is a smooth r-dimensional submanifold of X when Z contains a bielliptic curve section.O. Yasukura, in collaboration with H. Kaji (Waseda Univ., Japan and IMPA, Brasil), gave a concrete investigation on the relations among three objects : the adjoint varieties, symplectic triple systems and the gradation of contact type for complex simple Lie algebras. And they described and proved projective geometric properties on Freudenthal varieties in terms of the concept,of symplectic triple systems. In particular, for the adjoint varieties, the orbit decomposition and projective geometric description of the secant varieties are given. For Freudenthal varieties, the linear sectional relation with the corresponding adjoint varieties and an essential proof for the homogeneity are obtained as well as several other proves.