Analysis of J-holomorphic Curves

J全纯曲线分析

• 批准号: 9803554
• 负责人: Thomas Parker
• 金额: $ 6.33万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803554&HistoricalAwards=false
• 关键词: Analysis; holomorphic; Curves

AbstractProposal: DMS-9803554Principal Investigator: Thomas ParkerThis project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to develop effective methods forcomputing Gromov-Witten invariants of symplectic manifolds andenumerative invariants of algebraic manifolds. The main thrust is acontinuing project with E. Ionel on a gluing formula for Gromovinvariants under the operation of `symplectic connect sum'. The ideais to collapse the connect sum to a singular manifold, keeping trackof the holomorphic curves along the way. Parker and Ionel havealready obtained a gluing formula in special cases; these imply thewell-known recent formula of Caporaso-Harris. A general gluingformula should be an effective new tool for computing Gromov andenumerative invariants. A second part of the project seeks to foranswer the following question. Suppose C is curve that isJ-holomorphic for some non-generic J. If one perturbs J to a nearbygeneric J', how many J'-holomorphic curves are there close to C?Several well-known problems in enumerative geometry, some solved, someunsolved, reduce to this problem. Parker and Ionel have an approachbased on the `Taubes obstruction bundle' . The last part of theproject suggests using modified Gromov invariants to obtain invariantsthat count curves which exist only for special classes of almostcomplex structures.One of the most basic problems in mathematics is to determine thesolutions of a system of polynomial equations, and an important firststep toward that goal is to determine the NUMBER of solutions. Thereis an explicit formula for the number of simultaneous solutions of aset of n polynomials in n variables. One can then ask for the numberof solutions for n polynomials in n-1 variables. In this case thereis a free parameter, so the locus of solutions will be a union ofcurves. How many? This question has been systematically studied for100 years, but only a few special cases were solved. Then, around1990, it was realized that these problems can be translated intosymplectic geometry, and then tackeled using the powerful machinery ofmathematical gauge theory. (Gauge theory, originally part of physics,has been the focus of many very fruitful interactions betweenmathematicians and physicists over the past twenty years; it includesYang-Mills and Seiberg-Witten theory, and String theory). This`Gromov invariant' approach led quickly to formulas answering some ofthe original enumerative problems, and there are clear indicationsthat there are more to be discovered. This project is aimed towardfurther developing the symplectic gauge theory in order to produceadditional general formulas, and to meld these formulas into acoherent theory.

摘要项目提案：dms -9803554首席研究员：Thomas parker本项目涉及伪全纯曲线理论的解析方面。目的是建立计算辛流形的Gromov-Witten不变量和代数流形的枚举不变量的有效方法。主要内容是与E. Ionel继续研究在“辛连接和”操作下格罗莫不变量的粘接公式。其思想是将连接和折叠成奇异流形，并在此过程中跟踪全纯曲线。Parker和Ionel已经得到了特殊情况下的粘合公式；这暗示了最近著名的卡波拉索-哈里斯公式。一般胶合公式是计算Gromov不变量和枚举不变量的有效新工具。该项目的第二部分试图回答以下问题。假设C是一条曲线，对于某个非泛型J是J全纯的，如果将J扰动成一个近泛型J‘，那么在C附近有多少条J’全纯曲线？枚举几何中几个著名的问题，有的已经解决了，有的还没有解决，都归结为这个问题。Parker和Ionel提出了一种基于“Taubes阻塞束”的方法。项目的最后一部分建议使用修正的Gromov不变量来获得只存在于几乎复杂结构的特殊类别的曲线的不变量。数学中最基本的问题之一是确定多项式方程组的解，而实现这一目标的重要的第一步是确定解的数量。对于n个变量的n个多项式集的同时解的个数，有一个显式的公式。我们可以求n-1个变量的n个多项式的解的个数。在这种情况下，有一个自由参数，所以解的轨迹将是曲线的并集。有多少?这个问题已经系统地研究了100年，但只有少数特殊情况得到了解决。然后，在1990年左右，人们意识到这些问题可以转化为辛几何，然后利用数学规范理论的强大机制来解决。（规范理论原本是物理学的一部分，在过去的二十年里，它一直是数学家和物理学家之间许多卓有成效的互动的焦点；它包括yang - mills和Seiberg-Witten理论，以及弦理论）。这种“格罗莫夫不变量”的方法很快就得出了一些公式，可以回答一些最初的枚举问题，而且有明确的迹象表明，还有更多的问题有待发现。本项目旨在进一步发展辛规范理论，以产生更多的一般公式，并将这些公式融合到相干理论中。