轨形上的层理论及其在辛拓扑中的应用

The concept orbifold is a generalization of manifold, and plays an important role in the Gromov-Witten theory. Locally an orbifold looks like the quotient of an open subset of a vector space by a finite group. Due to historical causes, there are several versions of Gromov-Witten invariants for general closed symplectic manifolds:Fukaya-Ono's, Li-Tian's, Ruan's, Siebert's, Joyce's and etc. Among them, we know only a few relations. This is inconvenient for the applications of Gromov-Witten theory. So, in the project, we will focus on two of the versions: Fukaya-Ono's and Siebert's. By studying the sheaf theory on orbifold, especially the (co)homology groups with sheaf coefficients and duality properties between cohomology and homology groups, we will follow Siebert's ideas and reconstruct his localized Euler classes for oriented Kuranishi sections of Banach orbibundles in a special case: the base orbifold is just a point. Then we will compare (locally and then globally) Siebert's and Fukaya-Ono's virtual fundamental classes by using the properties of localized Euler classes. After doing this, we will obtain relations between Fukaya-Ono's and Siebert's symplectic Gromov-Witten invariants. We believe this project will be useful in the research of Gromov-Witten theory on closed symplectic manifolds and of sheaf theory on orbifolds or G-manifolds for some compact Lie group G.

轨形是流形的一种自然推广，也是Gromov-Witten 理论中常见的数学对象。由于历史原因，闭辛流形上的Gromov-Witten不变量定义出现了多种版本，而大部分版本之间的等价关系均未知，这为该理论的研究和应用带来了不便。为此，本项目拟以Fukaya-Ono版本和Siebert 版本定义为主要研究对象，通过研究轨形上的层理论特别是以任意层为系数的层(上)同调群和对偶关系，考察带有定向Kuranishi 截面的Banach 轨丛的局部化欧拉类的泛函性和相容性，先局部后整体地比较两种版本的虚基本类，从而得到这两种版本定义之间的关系。本项目旨在揭示Fukaya-Ono版本和Siebert 版本辛Gromo-Witten 不变量定义之间的显式关系以及利用局部卡语言描述轨形上的层理论，为完善辛Gromov-Witten 理论和研究轨形以及等变流形上的层理论提供前期工作准备。