Rigidity of local holomorphic maps between Hermitian symmetric spaces
Rigidity of local holomorphic maps between Hermitian symmetric spaces
复制标题
Hermitian对称空间间局部全纯映射的刚性
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Hanlong Fang
中科院分区：
文献类型：
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作者：
Hanlong Fang
本文研究了Hermite对称空间之间的局部全纯映射的刚性。对于从紧型的不可约Hermitian对称空间到自身的局部全纯映射，如果它具有一个标准Kähler-Schntein度量，我们证明了每个映射都扩张到流形的一个等距，只要这些映射满足一个测度保持方程并且是一般非退化的.为了建立刚性结果，在代数背景下引入了Serge簇和塞格雷族的概念.在得到主要定理之前，我们首先证明了一般情形下部分退化全纯映射的一个基本性质。然后，我们建立了Nash代数的这些映射之一，通过应用这个基本性质。在这里，流形到某个射影空间的模嵌入的显式本质上是使用。然后应用标准的单气味论证来证明这个纳什-代数映射的合理性。最后利用覆盖技巧证明了该映射是一个双有理映射，并进一步证明了它是一个等距映射。因此，通过归纳，我们得出了主要定理。本论文是在与黄晓军和肖明（[FHX]）合作的基础上完成的。
OF THE DISSERTATION Rigidity of local holomorphic maps between Hermitian Symmetric Spaces By HANLONG FANG Dissertation Director: Xiaojun Huang In this dessertaion, rigidity of local holomorphic maps between Hermitian symmetric spaces has been studied. For local holomorphic maps from an irreducible Hermitian symmetric spaces of compact type to itself, which is equipped with a canonical Kähler-Eisntein metric, we show that every map extends to an isometry of the manifold, provided that the maps satisfy a measure-preserving equation and are generically non-degenerate. To establish the rigidity result, a notion of Serge variety and Segre family in the algebraic setting is introduced. Before obtaining the main theorem, we first prove a basic property for partially degenerate holomorphic maps in a general setting. Then we establish the Nash-algebraicity for one of these maps by applying this basic property. Here the explicit expression of the mimimal embedding of the manifold into a certain projective space is essentially used. Standard monodormy argument is then applied to show the rationality for this Nash-algebraic map. Lastly by a covering trick we show that the map is a birational map and further an isometry. Hence by induction, we conclude the main theorem. This thesis is based on a joint work with Xiaojun Huang and Ming Xiao ( [FHX]).
影响因子：
1.7
作者：
Fang, Hanlong;Huang, Xiaojun;Xiao, Ming
通讯作者：
Xiao, Ming
DOI：
10.4310/ajm.2018.v22.n4.a7
发表时间：
2016-09
期刊：
arXiv: Complex Variables
影响因子：
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作者：
Ming Xiao;Yuan Yuan-Yuan
通讯作者：
Ming Xiao;Yuan Yuan-Yuan