Geometry and Topology of Counting Curves in Projective Manifolds

项目负责人：刘克峰项目将重点研究镜面对称的几何和拓扑，特别是在射影流形中计数曲线的方面。更确切地说，有三个突出而又密切相关的问题是我想要研究并希望解决的。首先证明了在任意投影流形及其Calabi-Yau子流形中计数有理曲线的最一般镜像原理；二是了解局部镜像对称的几何和代数，特别是应用镜像原理计算中出现的椭圆曲线所起的惊人作用；第三是建立了计算投影流形中高属曲线的镜像原理。我希望通过进一步扩展我们开发的技术来解决这些问题，以证明气球歧管的镜像原理。这三个问题是同一个问题的不同方面：以最一般的形式证明和理解镜像原理。超弦理论是科学中最具雄心的理论之一，它是为了实现自然规律的大统一而提出的。基于这一理论，弦理论家做出了许多非凡的数学猜想。镜像对称就是其中之一。这些猜想的证明不仅会给出漂亮的数学结果，而且有助于获得对物理理论的信心。我们发展的镜像原理部分地证实了其中的一些猜想。它有许多有趣的数学结果，并且与物理理论非常一致。

AbstractAward: DMS-0072182Principal Investigator: Kefeng LiuMy project will focus on the understanding of the geometry andtopology of mirror symmetry, in particular the aspect of countingcurves in projective manifolds. More precisely there are threeoutstanding and closely related problems that I would like towork on and hope to solve. The first is to prove the most generalmirror principle for counting rational curves in any projectivemanifold and its Calabi-Yau submanifolds; the second is tounderstand the geometry and algebra of the local mirror symmetry,in particular, of the surprising role played by the ellipticcurves appeared in our computation by applying mirror principle;the third is to develop a mirror principle for counting curves ofhigher genus in projective manifolds. I hope to solve theseproblems by further extending the techniques we developed toprove the mirror principle for balloon manifolds. These threeproblems are the different aspects of a single problem: to proveand understand the mirror principle in its most general form.Superstring theory, one of the most ambitious theory in sciences,is proposed for the grand unification of the laws of the nature.Based on this theory, string theorists have made many remarkablemathematical conjectures. Mirror symmetry is among one of them.The proofs of these conjectures will not only give beautifulmathematical results, but also help gain confidence in thephysical theory. The mirror principle we developed has partiallyverified some of these conjectures. It has many interestingmathematical consequences and is remarkably compatible with thephysical theory.