Smooth and Symplectic 4-Manifolds

光滑和辛 4 流形

• 批准号: 0072212
• 负责人: Ronald Fintushel
• 金额: $ 16.22万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072212&HistoricalAwards=false
• 关键词: Smooth; Symplectic; Manifolds

Project Title: Smooth and Symplectic 4-ManifoldsPI: Ronald FintushelAward: 0072212Abstract: The theory of smooth 4-manifolds gains its importance both from its central location between low and high-dimensional topology and from its close interaction with high energy physics. The major problem in this field is the classification of smooth simply connected4-manifolds. The interaction between topology and physics hasstimulated the construction of invariants - at first Donaldson's invariant, and then the invariant of Seiberg and Witten - which are useful in distinguishing the diffeomorphism types of 4-manifolds. These have led to major advances, and they have allowed workers to study new constructions of 4-manifolds. These have confused the issue of classification, but also have invigorated the theory and reinforced its richness. The theory is now left without even a conjectural classification. It seems that still further examples are needed to identify a suitable classification scheme, and the proposer intends to work on such constructions. One class of 4-manifolds which have exceptionally close ties to theoretical physics are those with a symplectic structure, and the last few years have also seen progress in the theory of symplectic 4-manifolds; especially new constructions, and most notably, Donaldson's work on Lefschetz fibrations. The proposer intends to continue work on the explicit constructions of Lefschetz fibrations and related questions on symplectic submanifolds. The ultimate goal of the proposer is to develop new constructions of smooth 4 manifolds in the hope that a general picture will begin to emerge. The focus of this project will be to construct new types of examples of smooth and symplectic 4-manifolds and to study the diversity of embedded symplectic submanifolds (up to smooth isotopy) in a given homology class. In particular, is every symplectic surface in the complex projective plane smoothly isotopic to a holomorphic curve? If one allows enough blowups, this is not true.Another issue is the geography problem for simply connectedirreducible 4-manifolds. Each such manifold can be assigned a lattice point in the plane corresponding to its characteristic numbers. The problem is to study which points are realized. There has been notableprogress, but much work still remains, and the principal investigator plans to seek new methods for constructing irreducible simply connected 4-manifolds of positive signature. These techniques are related to a kind of theory of minimal genus surfaces with constraints. Also he, along with R. Stern, conjectures a replacement for the Noether inequality for symplectic 4-manifolds, and they have a promising technique for its proof, which they plan to pursue.

项目名称：光滑和辛4-流形PI：罗纳德Fintushel奖项：0072212摘要： 光滑四维流形理论的重要性不仅在于它在低维和高维拓扑之间的中心位置，而且在于它与高能物理的密切相互作用。这一领域的主要问题是光滑单连通4-流形的分类。拓扑学与物理学的相互作用促进了不变量的构造--首先是唐纳森不变量，然后是Seiberg和维滕不变量--这些不变量在区分四维流形的单同态类型时非常有用。这些都导致了重大的进展，他们允许工人研究新的建设4流形。这些都混淆了分类问题，但也丰富了理论，加强了它的丰富性。这一理论现在甚至连理论分类都没有了。看来还需要更多的例子来确定一个适当的分类方案，提议者打算研究这种结构。一类4流形有非常密切的联系，理论物理是那些具有辛结构，并在过去几年中也看到了进展理论辛4流形;特别是新的建设，最值得注意的是，唐纳森的工作莱夫谢茨纤维。本文拟继续研究辛子流形上Lefschetz纤维化的显式构造及相关问题。提议者的最终目标是发展新的光滑4流形的构造，希望开始出现一个总体的图景。 这个项目的重点将是构造光滑和辛4-流形的新类型的例子，并研究在给定的同调类中嵌入的辛子流形（直到光滑合痕）的多样性。特别地，复射影平面上的每一个辛曲面是否光滑地与一条全纯曲线同构？另一个问题是单连通不可约四维流形的地理问题。每一个这样的流形可以被分配一个格点在平面上对应于它的特征数。问题是要研究哪些点是实现的。虽然取得了显著的进展，但仍有许多工作要做，主要研究者计划寻求新的方法来构造正签名的不可约单连通4-流形。这些技巧与一类带约束的极小亏格曲面理论有关。还有他，沿着还有R。Stern，为辛4-流形的Noether不等式提供了一个替代品，他们有一个很有前途的证明技术，他们计划继续下去。