On fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents
On fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents
复制标题
具有允许一维最小有理切线簇的 nef 切束的 fano 流形
DOI：
10.1090/s0002-9947-02-02953-7
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发表时间：
2002
影响因子：
1.3
通讯作者：
N. Mok
中科院分区：
文献类型：
--
作者：
N. Mok
设X是Picard数为1的具有数值有效切丛的Fano流形.根据Campana-Peternell猜想的主要情形，X应该双全纯于有理齐次流形G/P，其中G是单李群，P <$G是极大抛物子群。在我们看来，对于皮卡德数1的情形，没有压倒一切的证据证明坎帕纳-佩特内尔猜想在它的全部普遍性中是有效的。作为一般程序的一部分，作者已经进行了Jun-Muk黄通过其品种的最小合理的切线研究uniruled射影流形，一个新的几何方法是通过在当前的文章中的一个特殊情况下，包括（a）恢复的一般品种的最小合理的切线Cx，和（B）恢复结构的合理齐次流形从Cx。证明了当B 4（X）= 1且极小有理切线的一般簇为1维时，X与射影平面P2、三维超二次曲面Q3或G2型的5维Fano齐次切触流形K（G2）双全纯.主要的困难是方案的（a）部分。本文证明了CxCPTx（X）是一条次数小于3的有理曲线，并证明了d = 1.分别为2个3分别精确地对应于X = P2的情形。Q3分别K（G 2）。设κ是X上极小有理曲线的Chow分量的选择的归一化。切丛的负性意味着κ是光滑的。进一步地，它意味着在任意点x ∈ X，标记为的极小有理曲线的相应Chow空间的正规化κ x是光滑的。在证明了κ x是有理曲线之后，我们的主要研究对象是κ的泛族u，给出了一个双纤维化p：u → κ，μ：u → X，它给出了P1-丛。存在一个在κ上的秩为2的全纯向量丛V，它的射影化同构于p：u → κ。证明了V是稳定的，并由稳定性和Hermitian-Einstein度量的存在性定理所得到的不等式c21（V）< 4c 2（V）导出了不等式d < 4.在c21（V）= 4c 2（V）的特殊情况下，通过研究V上厄米-爱因斯坦度规的曲率张量的结构，排除了d = 4的情况.
Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C x , and (b) recovering the structure of a rational homogeneous manifold from C x . The author proves that, when b 4 (X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P 2 , the 3-dimensional hyperquadric Q 3 , or the 5-dimensional Fano homogeneous contact manifold of type G 2 , to be denoted by K(G 2 ). The principal difficulty is part (a) of the scheme. We prove that C x C PT x (X) is a rational curve of degrees < 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P 2 resp. Q 3 resp. K(G 2 ). Let κ be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that κ is smooth. Furthermore, it implies that at any point x ∈ X, the normalization κ x of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that κ x is a rational curve, our principal object of study is the universal family u of κ, giving a double fibration p: u → κ, μ: u → X, which gives P 1 -bundles. There is a rank-2 holomorphic vector bundle V on κ whose projectivization is isomorphic to p: u → κ. We prove that V is stable, and deduce the inequality d < 4 from the inequality c 2 1 (V) < 4c 2 (V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c 2 1 (V) = 4c 2 (V).