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维的复射影流形上的一个n-2阶的样本向量束，其截面的零轨迹Z是Kodaira维数为1的代数曲面。那么E的结构就完全确定了。这推广了Sommese和Shepherd-Barron关于样本除数的结果。(2)由维数为n的光滑复射影变数X和X上秩为n-1的充足向量束E构成的偏振变数（X， E）的分类，使得E有一个零轨迹为光滑椭圆曲线的截面。当E非常充裕，且截面上的零轨迹等于非小于2的超椭圆曲线时，研究了E的性质。(3)特别地，当Z的属等于2时，给出了这类（X， E）的分类。一种由维数为n的光滑复射影变数X和X上秩为n-r的充足向量束E组成的极化变数（X， E）的分类，使得当Z包含双椭圆曲线截面时，E有其零轨迹Z为X的光滑r维子流形的截面yasura与H. Kaji（日本早稻田大学和巴西IMPA）合作，对复单李代数的伴随变量、辛三重系统和接触类型的渐变这三个对象之间的关系进行了具体的研究。他们根据辛三重系统的概念，描述并证明了Freudenthal变体的射影几何性质。特别地，对于伴随变量，给出了割线变量的轨道分解和射影几何描述。对于Freudenthal变量，给出了其与相应伴随变量的线性截面关系和齐性的一个重要证明，并给出了其他几个证明。

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.