Gromov-Witten invariants of symplectic 4-manifolds

本项目旨在研究辛4-流形的Gromov-Witten不变量，特别关注Kahler曲面。它包括三个项目。PI（与托马斯H。帕克）证明了Kahler曲面（具有光滑正则因子）的GW理论可以简化为自旋曲线的局部GW理论。PI还表明，局部GW理论可以简化为零维相对局部GW理论。前两个项目的目标（与托马斯H。帕克）是完整地描述自旋曲线的零维相对定域GW理论。 这将给出Kahler曲面的GW不变量的完整计算（具有光滑正则因子）。第三个项目的目标是将Kahler曲面的GW不变量的结构定理扩展到一般的辛4-流形。这个扩展将提供一个合适的概念的分量的标准类，这是一个非常重要的辛不变量的拓扑辛4-流形的数量。近年来，新发展的GW不变量理论已经解决了一些长期存在的数学问题。这个理论提供了一个创新的方法来计算某些基本的几何对象称为全纯曲线。 由此产生的计数，以及计算它们的系统方法，已经在数学的两个经典领域：辛几何和枚举几何中取得了重大进展。GW不变量在数学和理论物理之间起着重要的桥梁作用。这个项目旨在开发新的几何技术计算GW不变量使用偏微分方程，拓扑学和代数几何的方法。在这个项目中开发的技术和想法应该在其他环境中非常有用。

This project aims to study the Gromov-Witten invariants of symplectic 4-manifolds, with special attention to Kahler surfaces. It consists of three projects. The PI (with Thomas H. Parker) showed that the GW theory of Kahler surfaces (with a smooth canonical divisor) can be reduced to the local GW theory of spin curves. The PI also showed that the local GW theory can be reduced to the dimension zero relative local GW theory. The goal of the first two projects (with Thomas H. Parker) is to completely describe the dimension zero relative local GW theory of spin curves. This will give a complete calculation of the GW invariants of Kahler surfaces (with a smooth canonical divisor). The goal of the third project is to extend aspects of a structure theorem for the GW invariants of Kahler surfaces to general symplectic 4-manifolds. This extension will provide a suitable notion of the number of the components of the canonical class which is a very important symplectic invariant for the topology of symplectic 4-manifolds. The extension will also remove the smoothness condition of canonical divisor in the previous projects.In recent years, some long-standing mathematics problems have been solved using the newly-developed theory of GW invariants. This theory provides an innovative method for counting certain fundamental geometric objects called holomorphic curves. The resulting counts, and the systematic methods for calculating them, have spurred significant advances in two classical fields of mathematics: symplectic geometry and enumerative geometry. The GW invariants also play an important role bridging mathematics and theoretical physics. This project seeks to develop new geometric techniques for calculating GW invariants using methods in partial differential equations, topology and algebraic geometry. The techniques and ideas developed in this project should be very useful in other contexts.