The Geometry of Flag Manifolds
The Geometry of Flag Manifolds
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旗形流形的几何形状
DOI：
10.1112/plms/s3-9.2.253
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发表时间：
1959
影响因子：
1.8
通讯作者：
D. Monk
中科院分区：
文献类型：
--
作者：
D. Monk
射影空间S_n中的一个标志是子空间的一个‘巢’，每个子空间的一个维度是从0到n-1。所有这些标志的集合与一个代数簇的点是一致的，即空间的标志流形，记为F(n-\*)。流形可以表示为齐次空间，它是酉群U{n-{-\)的极大环面的商，Ehresmann在他的论文(6)中从这个角度考虑了它。近年来，齐次空间的理论得到了较大的发展，Borel(1)对齐次空间的拓扑学作了详细的研究。在(2)中可以找到一个没有证据的很好的调查。本文的工作是通过比较Ehresmann和Borel的结果而提出的。它的目的是主要用经典代数几何的方法研究旗流形的几何性质，并证明这些作者所得到的不同形式的基之间的关系。本文的大部分内容是关于交点公式的表述和证明。在此之前，霍奇的&连通性理论被用来计算流形的公设和阶，而Ehresmann的基记法被简化为一种不仅更容易写而且自然地符合现代理论框架的形式。一旦证明了交集公式，两个基之间的联系就很快建立起来了。JP(3)是射影平面的入射点线元素的流形，是S7中的VI，是两个平面的Segre积的素数截面。这变种是众所周知的，已经被几个作者研究过，包括Semple(16)和Severi(18)。更高空间的旗形流形似乎没有在很大程度上被几何处理，也许是因为它们的阶数和维度高得令人望而却步，正如定理1所示。然而，在(14)中，Longo考虑了一类包含F(4)的变种作为特殊情况。像桑普尔一样，他主要关注的是微分曲线元素，而不是旗帜流形本身。Martinelli(15)研究了一个更一般的“不完全旗子流形”的简单情况(见§2)，通过一种有趣的归纳方法获得了基数。他使用了一种记号，
Aflag in projective space Sn is a'nest'of subspaces, one of each dimension from 0 to n—1. The aggregate of all such flags is in unexceptional birational correspondence with the points of an algebraic variety, the flag manifold of the space, denoted by F (n-\*\). The manifold can be represented as a homogeneous space, being the quotient of the unitary group U {n-{-\) by a maximal torus, and from this point of view it was considered by Ehresmann in his thesis (6). In recent years the theory of homogeneous spaces has been extended considerably, and Borel (1) has made a detailed study of their topology. An excellent survey, without proofs, is to be found in (2). The present work was suggested by a comparison of the results of Ehresmann and Borel. Its purpose is to study the geometrical properties of the flag manifold, mainly by the methods of classical algebraic geometry, and to exhibit the relation between the different forms of the basis obtained by these authors. A large part of the paper is devoted to the formulation and proof of an intersection formula. In preparation for this, Hodge's theory of &-connexes is used to calculate the postulation and order of the manifold, and Ehresmann's notation for the basis is simplified to a form which is not only more convenient to write but which fits naturally into the framework of the modern theory. Once the intersection formula has been proved, the connexion between the two bases is quickly established. The paper concludes with a brief statement of some unsolved problems.JP (3), the manifold representing incident point-line elements of the projective plane, is a VI in S7, being a prime section of the Segre product of two planes. This variety is well known, having been studied by several authors, including Semple (16) and Severi (18). The flag manifolds of higher spaces do not appear to have been treated geometrically to any great extent, perhaps because their orders and dimensions are prohibitively high, as a glance at Theorem 1 will show. However, in (14), Longo has considered a class of varieties which includes F (4) as a particular case. Like Semple he is concerned principally with differential curve elements, rather than with the flag manifolds themselves. Martinelli (15) has investigated a simple case of the more general'manifold of incomplete flags'(see § 2), obtaining the basis by an interesting induction method. He uses a notation,