Gromov-Witten invariants of symplectic 4-manifolds

本课题旨在研究辛4流形的Gromov-Witten不变量，特别关注Kahler曲面。它由三个项目组成。PI（与Thomas H. Parker）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的相对局部GW理论。前两个项目（与Thomas H. Parker合作）的目标是完全描述自旋曲线的零维相对局部GW理论。这将给出Kahler曲面的GW不变量的完整计算（具有光滑正则除数）。第三个项目的目标是将Kahler曲面的GW不变量的结构定理的各个方面扩展到一般辛4流形。该扩展将提供一个合适的正则类分量数的概念，这是辛4流形拓扑中一个非常重要的辛不变量。该扩展还将删除以往项目中正则除数的平滑条件。近年来，利用新发展的GW不变量理论解决了一些长期存在的数学问题。这个理论提供了一种创新的方法来计算某些称为全纯曲线的基本几何对象。由此产生的计数，以及计算计数的系统方法，推动了两个经典数学领域的重大进展：辛几何和枚举几何。GW不变量也在数学和理论物理之间起着重要的桥梁作用。该项目旨在开发新的几何技术，利用偏微分方程、拓扑和代数几何中的方法计算GW不变量。在这个项目中开发的技术和想法在其他情况下应该非常有用。

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.