Stability of tangent bundles of low dimensional Fano manifolds with Picard number 1
Stability of tangent bundles of low dimensional Fano manifolds with Picard number 1
复制标题
皮卡德数1的低维Fano流形切丛的稳定性
DOI：
10.1007/s002080050237
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发表时间：
1998
影响因子：
1.4
通讯作者：
Jun
中科院分区：
文献类型：
--
作者：
Jun
受Fano流形上关于Kähler-Einstein度量的Calabi问题的启发，许多作者研究了Fano流形的切丛何时稳定的问题。在[PW]中，研究了Picard数为1的Fano流形，并证明了它们中的许多流形具有稳定的切丛。特别地，证明了所有Picard数为1且维数≤ 4的Fano流形都有稳定切丛.它们的证明依赖于上同调消元和大指数Fano流形分类理论的结果，本文将介绍一种基于有理曲线的不同方法，该方法相当容易地给出Picard数为1的Fano流形的切丛的（半）稳定性.有理曲线与切丛稳定性之间的关系最初由Miyaoka在他关于非uniruled簇的切丛的通有半负性的工作中使用（[Mi]，[Sh]）。在这里，我们进一步推动和研究的关系，以射影几何的各种切线方向的有理曲线。我们从有理曲线稳定性定义的重新表述开始。这是[OSS]中定义的简单概括。设X是Picard数为1的n维Fano流形，T是X的切丛.固定Chow方案的一个分支K，它的类属成员对应于X上的有理曲线，其中T的限制分裂为O（2）<$[O（1）] p <$O n− 1− p，其中p< n。回想一下，这样的K总是通过[Mr]的弯折论证而存在。众所周知，
Motivated by the Calabi problem concerning Kähler-Einstein metrics on Fano manifolds, the problem when the tangent bundle of a Fano manifold is stable has been studied by many authors. In [PW], Fano manifolds with Picard number 1 were studied and it was proved that many of them have stable tangent bundles. In particular, it was proved that all Fano manifolds with Picard number 1 of dimension≤ 4 have stable tangent bundles. Their proof depends on some results on cohomological vanishing and on the classification theory of Fano manifolds of large index.In this note, we would like to introduce a different approach, based on rational curves, which gives the (semi-) stability of the tangent bundles of some Fano manifolds with Picard number 1 rather easily. The relation between rational curves and the stability of tangent bundles was originally used by Miyaoka in his work on the generic seminegativity of the tangent bundles of non-uniruled varieties ([Mi],[Sh]). Here we push it further and study the relation to the projective geometry of the variety of tangent directions to rational curves. We start with a reformulation of the definition of the stability in terms of rational curves. This is a straightforward generalization of the definition in [OSS]. Let X be an n-dimensional Fano manifold with Picard number 1 and T be the tangent bundle of X. Fix a component K of the Chow scheme whose generic members correspond to rational curves on X, where the restriction of T splits as O (2)⊕[O (1)] p⊕ O n− 1− p for some integer p< n. Recall that such K always exists by the bend-and-break argument of [Mr]. Also it is well-known that given