局部 Hermite 对称空间的复子流形

In this project we study complex submanifolds of locally Hermitian symmetric spaces, basing on the conic structures or conic connections of the ambient space and the holomorphic tangent sequence of the complex submanifold. We will focus on the following problems: (1) Characterization of special submanifolds such as totally geodesic submanifolds; (2) Inheritance of conic structures or conic connections on submanifolds; (3) Rigidity of conic sub-structures. . Locally Hermitian symmetric spaces are the moduli spaces of many important mathematical objects, and therefore the characterization of special submanifolds of these spaces has attracted a lot of attention from different fields in mathematics. We will focus on those complex submanifolds whose holomorphic tangent sequence splits, aiming at extending the known results for complex space forms to locally Hermitian symmetric spaces of rank at least 2. On the other hand, the tangent directions given by minimal rational curves or minimal disks form a special geometric structure on locally Hermitian symmetric spaces. This geometric structure is a very important special case of the so-called cone structure in holomorphic geometry. In this project we will also study how this geometric structure interacts with the complex submanifolds, first aiming at giving conditions for the inheritance of such structure on complex submanifolds, and then investigating the recognition and rigidity problems of such special submanifolds.

本项目拟以局部Hermite对称空间上经典几何机构，特别是锥结构与锥联络，为出发点，透过考虑复子流形上的全纯切丛序列，研究对称空间里子流形的刻画和刚性问题。研究的课题包括：（1）全测地子流形等特殊子流形之刻画；（2）锥结构或锥联络在子流形上之承继；（3）子锥结构之刚性问题。. 作为很多重要数学对象的模空间，局部Hermite对称空间里如何刻画特殊子流形一直也是备受关注的问题，本项目拟聚焦于那些拥有分裂全纯切丛序列的复子流形，推广复空间形式（秩为1）中已知的结果到秩至少为2的局部Hermite对称空间。另一方面，局部Hermite对称空间拥有丰富的几何结构，为空间中极小有理曲线或极小单圆盘在切丛上所给出的方向生成，是全纯几何里所谓锥结构的一个重要特例。本项目拟研究对称空间上这些结构与子流形的相互影响，先给出子流形能承继这些结构的条件，后探讨这些特别子流形的各种刻画与刚性问题。

设 X 为一个复流形，而 S 为 X 中一个紧致复子流形。S 称为 X 的分裂复子流形，若 X 的全纯切丛限制在 S 上时全纯同构与 S 的全纯切丛与 S 的法丛的直和。此项目研究的是当 X 为非紧型 Hermite 局部对称空间时，其分裂复子流形的刚性问题。当 X 的秩至少为 2 的时候，我们找出一个具体的下界（跟 X 有关），使得当 S 为分裂子流形且维数大于该下界时，则 S 为全测地子流形。这个结果对全测地性子流形（本为一几何对象）给出了一个纯分析性的刻画。