Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles
Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles
复制标题
皮卡德数 1 的 Fano 流形上的有限态射，具有带有平凡法丛的有理曲线
DOI：
10.1090/s1056-3911-03-00319-9
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发表时间：
2003
影响因子：
1.8
通讯作者：
N. Mok
中科院分区：
文献类型：
--
作者：
Jun;N. Mok
设X是Picard数为1的Fano流形，具有平凡法丛的有理曲线，f：X‘→X是从射影流形X’到X的一般有限满射全纯映射.当区域流形X‘是固定的，且目标流形X是先验允许变形的时，我们证明了全纯映射f：X’→X是局部刚性的，直至目标流形的双全纯.这一结果用一种完全不同的证明方法补充了我们以前的一个局部刚性定理(见J.Math。普雷斯应用。80(2001)，563-575)，对于目标流形X是Picard数为1的Fano流形，其上没有具有平凡法丛的有理曲线的类似情况。在另一个方向，给定一个Picard数为1的Fano流形X‘，我们证明了X’到Fano流形(Picard数为1的必要条件)上的广义有限满射全纯映射的一个有限结果，该结果允许具有平凡法丛的有理曲线.因此，任何Picard数为1的三维Fano流形只能支配有限个射影的同构类
Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → X be a generically finite surjective holomorphic map from a projective manifold X′ onto X. When the domain manifold X′ is fixed and the target manifold X is a priori allowed to deform we prove that the holomorphic map f : X′ → X is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563– 575) for the analogous situation where the target manifold X is a Fano manifold of Picard number 1 on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold X′ of Picard number 1, we prove a finiteness result for generically finite surjective holomorphic maps of X′ onto Fano manifolds (necessarily of Picard number 1) admitting rational curves with trivial normal bundles. As a consequence, any 3-dimensional Fano manifold of Picard number 1 can only dominate a finite number of isomorphism classes of projective