Topology of Sympletic Manifolds

当前研究项目的目标是研究一个torus manifold和manifold的变量之间的关系。在研究的课程中，我们成功地发展了一个通用理论的多粉丝组合学，并以一种形式应用于torus歧管的拓扑问题。在这个意义上，原来的目标被抓住了。主要结果将在以下情况下进行统计：1. We defined a notion of TイイD2 genus of a multi-fan， and showed that it coincides with the ordinary TイD2 genus of torus manifolds for multi-fans of the torus manifolds。We further showed an equality concerning the T-y-D2 genus similar to the one which holds between so-called h-vectors and f-vectors in combinatorics。我们的公式可能是为了给旧的术语一个新的解释。我们引入了添加剂中的多聚乙烯醇的运动，定义了双聚酯-海克曼函数和多聚乙烯醇的限定数字。它展示了双胞胎-海克曼的功能和每个不同的数字。这一般是一个已知的结果，与多聚物相关的torus manifolds。我们还将多功能公式的通用化作为多功能者的案例。使用双绞线-Heckman函数作为一个多粉丝的通用化的Ehrhart polynomial已经出现。我们展示了与多聚物柱的最高学位学期的协同作用，与多聚物柱的恒定学期协同作用，与多聚物的托德基因共同作用。《埃哈特政治的二元论》也是如此。Ehrhart polynomial of a convex polynomial与Riemann-Roch number of the对应的示例线捆绑的相对变体密切相关。根据命令，将这种现象一般化为多粉丝和多聚物，我们定义了等价的同源性和Gysin同源性，并获得了Ehrhart多聚物的同源性公式。

The aim of the present research project was to study the relation between the structure of the multi-fan of a torus manifold and invariants of the manifold. In the course of research we succeeded in developing a general theory of combinatorics of multi-fans in a form suited to be applied to topological problems of torus manifolds. In this sense the original aim was attained. Main results will be stated in the following.1. We defined a notion of TィイD2yィエD2 genus of a multi-fan, and showed that it coincides with the ordinary TィイD2yィエD2 genus of torus manifolds for multi-fans of the torus manifolds. We further showed an equality concerning the TィイD2yィエD2 genus similar to the one which holds between so-called h-vectors and f-vectors in combinatorics. Our formulation might be considered to give a new interpretation for the old quation.2. We introduced the notion of multi-polytope in addition, and defined the Duistermaat-Heckman function and the winding number for multi-polytope. It was shown that the Duistermaat-Heckman function and the winding number determined each other. This generalizes a result known for multi-polytopes associated to torus manifolds. We also gave a generalization of multiplicity formula to the case of multi-fans.3. Using the Duistermaat-Heckman function of a multi-fan a generalization of the Ehrhart polynomial is obtained. We showed the coefficient of the highest degree term coincided with the column of the multi-polytope and the constant term coincided with the Todd genus of the multi-polytope. The duality of the Ehrhart polynomial was also shown.4. The Ehrhart polynomial of a convex polytope is closely related to the Riemann-Roch number of the corresponding ample line bundle over the relevant toric variety. In order to generalize this phenomenon to the case of multi-fans and multi-polytopes we defined the equivariant cohomology and Gysin homomorphism, and obtained a cohomological formula of Ehrhart polynomial.