李代数上复结构的形变问题的研究——关于形变等式的研究

Research on Deformation of Complex Structures on Lie Algebras—Research on Deformation Equations

摘要: 在这篇文章中，我们将考虑李代数上的复结构的形变，主要是与复流形的情况作类比，我们期待会有一些新的现象出现。此外，预期目标：为李代数上复结构的形变建立一个完整的理论，并期望在这个理论的基础上，进一步加深关于李代数上复结构的形变与复流形的形变之间联系的理解。我们引入Banach不动点定理(压缩映像原理)作为新方法解决了李代数上的形变问题，并给出相关结论的新证明，这一证明极大地简化了Kodaira-Spencer的工作，更给了我们在做形变问题时的新视角。

Abstract: In this paper, we will consider the deformation of complex structures on Lie algebras, mainly by analogy with the case of complex manifolds. We expect some new phenomena to emerge. In addition, the expected goal is to establish a complete theory of the deformation of complex structures on Lie algebras, and to further deepen the understanding of the relation between the deformation of complex structures on Lie algebras and the deformation of complex manifolds on the basis of this theory. We introduce Banach fixed point theorem (compressed image principle) as a new method to solve the deformation problem on complex manifold, and give a new proof of related conclusions, which greatly simplifies Kodaira-Spencer’s work, and gives us a new perspective on deformation problems.