【作者】 陈浩；

【导师】 王奎；

【作者基本信息】 苏州大学 ， 基础数学， 2021， 硕士

【摘要】 本文的主要讨论对象是有n+1个离散不动点的有S1哈密顿作用的2n维紧致辛流形.首先介绍了辛流形上辛形式的基本概念和性质,并对常见的流形(如R2n,S2n,RPn,CPn)是否是辛流形作了探讨.在此基础上,本文进一步介绍了辛流形上的辛向量场和哈密顿向量场的概念和性质,然后定义了一般李群在辛流形上的哈密顿作用和流形上的矩映射,并证明了二维紧致辛流形中只有S2上有哈密顿作用.事实上,哈密顿作用的不动点和哈密顿向量场的零点及矩映射上的临界点是相同的.为了探究离散的不动点,接下来介绍了 Morse理论和等变上同调理论这两个重要的工具.对于有S1哈密顿作用的2n维紧致辛流形,不动点一定是存在的,作用的矩映射是流形上的完美的Morse函数,Morse指数为2i的不动点和辛流形的第2i阶上同调群之间存在着一定的对应关系,由此可以证明2n维辛流形上的哈密顿作用的离散的不动点至少有n+1个(命题4.2).一个重要的例子是S1在CPn上的哈密顿作用恰好只有n+1个离散的不动点.最后本文假设哈密顿作用的离散的不动点只有n+1个,此时不动点处的Morse指数是不动点处负权数的个数的两倍,并用Morse定理揭示了和这样的辛流形同伦等价的有限胞腔复形.进一步地,当辛流形的上同调环和CPn的上同调环同构时,利用等变上同调理论分别求得不动点处正负权数的乘积(命题4.4),并在辛流形是二维和四维的情况下,对这些权数的乘积做了分解,求得出具体的权数和S1在CPn上作用的不动点处的权数是一致的.更多还原

【Abstract】 This paper is mainly about compact symplectic manifolds of dimension 2n with S1 Hamiltonian action and n+1 isolated fixed points.At first,we introduce the concept and properties of the symplectic structure on symplectic manifolds and take R2n,S2,CPn as examples.With that,we give the definitions of symplectic vector field and Hamiltonian vector field and provide information about the Hamiltonian Lie actions and momemt map,proved that only S2 has S1 Hamiltonian action in the category of two-dimensionality.In fact,Hamiltonian action’s fixed points,Hamiltonian vector field’s zero points and moment map’s critical points are the same.In order to study the isolated fixed points,we introduce the Morse theory and S1 equivariant cohomology in the next part.For the 2n-dimensional compact symplectic manifolds with S1 Hamiltonian action,the fixed point exists.So there are relationship between the fixed point and manifold’s cohomology group.Then we prove such actions have at least n+1 fixed points(Proposition 4.2).An important example is that the S1 Hamiltonian action on CPn has exactly n+1 fixed points.At last,we suppose such action just has n+1 fixed points.So the morse index of the fixed point is twice as the count of negative weights on the fiexd point.By the Morse theory,the manifold has the homotopy type of a CW complex.When the integral cohomology ring of M is isomorphic to that of CPn,we use equi variant cohomology theory to get the product of the negative weights and that of the positive weights for every fixed point(Proposition 4.4).When manifold’s dimension are two or four,the product should have factorization.Therefore we can get all weights which is the same form as CPn’s weights behave.更多还原