Fano contact manifolds and nilpotent orbits
Fano contact manifolds and nilpotent orbits
复制标题
Fano 接触流形和幂零轨道
DOI：
10.1007/s000140050069
复制
发表时间：
1997
影响因子：
0.9
通讯作者：
A. Beauville
中科院分区：
文献类型：
--
作者：
A. Beauville
抽象的。复流形 M 上的接触结构是 TM 的 Corrank 1 子丛 F，使得 F 上的双线性形式（其值位于从向量场的李括号推导出来的商线丛 L = TM/F 中）处处非简并。在本文中，我们考虑 M 是 Fano 流形的情况；这意味着 L 是充足的。¶如果
${\frak g}$ 是一个简单的李代数，唯一的闭轨道
${\bold P}({\frak g})$（对于伴随作用）是 Fano 接触流形；据推测，每个 Fano 接触流形都是通过这种方式获得的。肯定的答案意味着具有正标量曲率的紧致四元数-卡勒流形的类似结果，这是黎曼几何中长期存在的问题。¶在本文中，我们在附加假设下解决了猜想，即 M 的接触自同构群是还原性的，并且有理图 M 的图像
与 L 相关的 $--\rightarrow$P(H0(M, L)*) 具有最大维度。证明依赖于半简单李代数中幂零轨道的性质，特别是 R. Brylinski 和 B. Kostant 的工作。
Abstract. A contact structure on a complex manifold M is a corank 1 subbundle F of TM such that the bilinear form on F with values in the quotient line bundle L = TM/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.¶If
${\frak g}$ is a simple Lie algebra, the unique closed orbit in
${\bold P}({\frak g})$ (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.¶In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M
$--\rightarrow$P(H0(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant.