Defect relations for holomorphic maps between spaces of different dimensions
Defect relations for holomorphic maps between spaces of different dimensions
复制标题
不同维度空间之间的全纯映射的缺陷关系
DOI：
10.1215/s0012-7094-87-05512-8
复制
发表时间：
1987
影响因子：
2.5
通讯作者：
Y. Siu
中科院分区：
文献类型：
--
作者：
Y. Siu
现在有一个相当丰富的理论亏损关系的全纯映射之间的空间，相等的维数开发的Nevanlinna，斯托尔，格里菲斯和许多其他人。然而，对于不等维空间之间的映射的情况，当目标空间是复射影空间时，基本上只有Weyl-Weyl和Ahlfors发展的理论，并且他们的理论建立在射影空间的射影线性结构上。这些结果的部分参考文献列表在本文的结尾。本文导出了不同维数空间之间全纯映射的亏损关系，其中目标空间可以是任意紧复代数流形。由于从C到复维度为2的紧致复流形的映射的特殊情况已经包含了一般情况所需的方法和技术，为了避免不必要的复杂符号并使论点更容易理解，我们仅给出完整的细节对于这种特殊情况，并指出一般情况需要的简单修改。除了利用Poincar ~ 6-Lelong公式、由被积函数的积分估计被积函数的微积分引理和对数的导数等值分布理论中常用的技巧外，本文的一个新的关键点是引入一个亚纯联络，使得除数关于该联络的“第二基本形式”为零.更精确地说，对于从C到紧Kihler曲面M的亚纯映射f，为了考虑正全纯线丛L的全纯截面s的非奇异零点集的亏损，我们引入了一个具有以下两个性质的亚纯联络I '，v/ forM：（i）对于M上的某个Hermitian全纯线丛F和F的某个全纯截面0，
There is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by Nevanlinna, Stoll, Griffiths and many other people. However, for the case of maps between spaces of unequal dimension essentially there is only the theory developed by Weyl-Weyl and Ahlfors when the target space is the complex projective space, and their theory is built on the projective linear structure of the projective space. A partial list of references for these results is given at the end of this paper. In this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold. Since the special case of maps from C to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case, in order to avoid unnecessarily complicated notations and to make the arguments more easily understood, we give complete details only for this special case and indicate the easy modifications needed for the general case. Besides the usual techniques in value distribution theory of using the Poincar6-Lelong formula, the calculus lemma of estimating the integrand by its integral, and the concavity of logarithm, a new key point in our argument is to introduce a meromorphic connection so that the "second fundamental form" of a divisor with respect to the connection is zero. More precisely, for the case of a meromorphic map f from C to a compact Kihler surface M, to consider the defect for a nonsingular zero-set of a holomorphic section s of a positive holomorphic line bundle L we introduce a meromorphic connection I’,v/ forM with the following two properties: (i) For some Hermitian holomorphic line bundle F over M and some holomorphic section 0 of F,