四维流形上局部线性和辛群作用的性质研究

In this project, we will apply the important achievement of gauge field theory, such as Seiberg-Witten theory and G-equivariant Seiberg-Witten-Taubes theory, associated the algebraic representation theory, to study the different properties of four-manifolds endowed different structures with different group actions as local linear actions and symplectic actions. On one hand, we research the automorphisms of some four -manifolds, such as the E8⊕E8 manifold, using G - Signature formula consider the representation and properties of the fixed points under the local linear actions as well as the realization problem. On the other hand, we focus the research on the category of symplectic four-manifolds. Using G-equivariant Seiberg - Witten - Taubes theory, we would like to improve the resuls of the rigidity theorem of symplectic elliptic surfaces under homologically trivial symplectic group actions with constraints, hoping to extend the results to more general situations. We also make effort for other symplectic four-manifolds with other symmetries under homologically nontrival actions, considering the representations and properties of fixed points, to supplement the answer to the the problem of rigid property.

本项目拟应用规范场论的重大成果Seiberg-Witten理论和G-等变Seiberg-Witten-Taubes的理论，结合代数表示论对加载不同的结构的四维流形上局部线性和辛群作用进行研究。一方面对一些四维流形上自同构群进行研究，如E8⊕E8等流形，利用G-Signature公式考虑其上局部线性作用的不动点表示和性质以及实现问题；另一方面在辛四维流形的范畴上进行研究，利用G-等变Seiberg-Witten-Taubes的理论，对辛椭圆曲面上带限制条件的同调平凡群作用的刚性定理进行改进，希望将结果扩展到更一般的情形中去；同时也尝试对非椭圆曲面的辛四维流形上的其他辛对称进行研究，考虑非同调平凡的群作用下的表示和不动点结构，以补充解答辛四维流形辛群作用的刚性性质的问题。