Studies on Pseudo-holomorphic Maps

DMS-0104331摘要本项目涉及伪全纯曲线理论的分析方面。目的是发展计算辛流形的Gromov-Witten不变量和代数流形的计数不变量的有效方法。这项工作建立在P.I.S最近与E.Ionel关于GW不变量的‘辛和公式’的工作的基础上。第一个项目涉及将求和公式推广到P.I.S学生李俊浩最近定义的“修正的GW不变量”。这将使该公式适用于具有$p_G0$的K“Ahler曲面，其中存在GW方法目前无法接近的重要猜想。第二个项目是一种辛方法，用来理解物理学家对计算Calabi-Yau三重曲线的母函数的预测。其目的是通过采用C·陶布斯发展的一些分析方法来证明预测公式，这些方法是为了将Seiberg-Witten和Gromov不变量联系起来。最后两个项目也与辛和公式有关。人们寻找用通常的GW不变量以及$X$和$V$的后代类来表示$(X，V)$对的相对Gromov-Witten不变量的公式。另一种是将求和公式推广到更一般类型的辛和，这是在代数几何中，当人们考虑射影线性系统时出现的。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目。一组n元n次多项式同时解的个数有一个显式公式。然后可以求出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，因此解的轨迹将是曲线的并集。多少?对这一问题的系统研究已有100年之久，但只解决了少数特例。然后，大约在1990年，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。(规范理论最初是物理学的一部分，在过去20年里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；它包括杨-米尔斯和塞伯格-威腾理论，以及弦理论)。这种“格罗莫夫-威腾不变量”的方法很快就产生了回答一些最初的计数问题的公式，而且有明显的迹象表明还有更多的问题有待发现。这个项目的目的是为了进一步发展辛规范理论，以便产生更多的通用公式，并将这些公式融合为一个连贯的理论。

Abstract for DMS - 0104331This project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to developeffective methods for computing Gromov-Witten invariants of symplectic manifolds and enumerative invariants of algebraic manifolds. This work builds on the P.I.'s recent work with E. Ionel on the `sympletic sum formula' for GW invariants. The first project involves extending the sum formula to the `modified GW invariants' recently defined by the P.I.'s student Junho Lee. This would make the formula applicable to K\"ahler surfaces with $p_g0$ where there areimportant conjectures that are not currently approachable by GWmethods. The second project is a symplectic approach to understanding physicists' predictions about the generating functions which count curves in Calabi-Yau 3-folds. The goal is to prove the predicted formulas by adapting some analytic methods C. Taubes developed to relate the Seiberg-Witten and Gromov invariants. The last two projects also relate to the sympletic sum formula. One seeks formulas expressing the relative Gromov-Witten invariants of a pair $(X,V)$ in terms of theusual GW invariants and the descendant classes of $X$ and $V$. The other proposes extending the sum formula to a more general type of symplectic sum which occurs in algebraic geometry when one considers projective linear systems.One of the most basic problems in mathematics is to determine the solutions of a system of polynomial equations, and an important first step toward that goal is to determine the NUMBER of solutions. There is an explicit formula for the number of simultaneous solutions of a set of n polynomials in n variables. One can then ask for the number of solutions for n polynomials in n-1 variables. In this case there is a free parameter, so the locus of solutions will be a union of curves. How many? This question has been systematically studied for 100 years, but only a few special caseswere solved. Then, around 1990, it was realized that these problems can be translated into symplectic geometry, and then tackeled using the powerful machinery of mathematical gauge theory. (Gauge theory, originally part of physics, has been the focus of many very fruitful interactions between mathematicians and physicists over the past twenty years; it includes Yang-Mills and Seiberg-Witten theory, and String theory). This `Gromov-Witten invariant' approach led quickly to formulas answering some of the original enumerative problems, and there are clear indications that there are more to be discovered. This project is aimed toward further developing the symplectic gauge theory in order to produceadditional general formulas and to meld these formulas into a coherent theory.