Gromov-Witten invariants of symplectic 4-manifolds

辛 4 流形的 Gromov-Witten 不变量

• 批准号: 1206192
• 负责人: Junho Lee
• 金额: $ 10.24万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206192&HistoricalAwards=false
• 关键词: Gromov; Witten; invariants; symplectic; manifolds

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.

本项目旨在研究辛4-流形的Gromov-Witten不变量，特别是Kahler曲面。它由三个项目组成。PI(与Thomas H.Parker)表明，Kahler曲面的GW理论(具有光滑的正则因子)可以归结为局部的GW自旋曲线理论。PI还表明，局部GW理论可以归结为零维相对局部GW理论。前两个项目(与Thomas H.Parker)的目标是完整地描述自旋曲线的零维相对局域GW理论。这将给出一个完整的计算Kahler曲面的GW不变量(具有光滑的正则因子)。第三个项目的目的是将Kahler曲面的GW不变量的结构定理的一个方面推广到一般的辛四维流形。这一推广将为正则类的分量的个数提供一个合适的概念，正则类是辛4-流形拓扑的一个非常重要的辛不变量。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。这一理论为计算称为全纯曲线的某些基本几何对象提供了一种创新的方法。由此产生的计数和计算它们的系统方法，促进了数学的两个经典领域-辛几何和计数几何-的重大进步。GW不变量也在数学和理论物理之间起着重要的桥梁作用。这个项目寻求开发新的几何技术来使用偏微分方程组、拓扑学和代数几何中的方法来计算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。