Research on homogeneous projective varieties by Lie algebra and algebraic geometry

H. Maeda给出了以下结果：（1）设E是n维复射影流形上的n-2秩的充分向量丛，其零轨迹Z是1维科代拉代数曲面。从而完全确定了E的结构。这推广了Sommese和Shepherd-Barron关于充分因子的结果。(2)极化簇（X，E）的一种分类，由一个n维光滑复射影簇X和一个在X上秩为n-1的样本向量丛E组成，使得E有一个零轨迹为光滑椭圆曲线的截面。并研究了当E很充分地具有零轨迹等于亏格不小于2的超椭圆曲线的截面时E的性质。(3)特别地，当Z的亏格等于2时，给出了这类（X，E）的分类.极化簇（X，E）的一种分类，它由一个n维光滑复射影簇X和一个在X上秩为n-r的充分向量丛E组成，使得E有一个截面，当Z包含一个双椭圆曲线截面时，其零轨迹Z是X的一个光滑r维子流形O. Yasukura与H. Kaji（早稻田大学，Japan和IMPA，Brasil）对复单李代数的伴随簇、辛三元系和切触型的阶化这三个对象之间的关系进行了具体的研究。他们用辛三系的概念描述和证明了Freudenthal簇上的射影几何性质。特别地，对于伴随簇，给出了割线簇的轨道分解和射影几何描述。对于Freudenthal簇，得到了它与相应伴随簇的线性截关系和齐性的一个本质证明，以及其它几个证明。

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.