Smooth and Symplectic 4-Manifolds

项目标题：光滑和辛4-流形PI：Ronald Fintushel奖：0072212摘要：光滑4-流形理论因其在低维和高维拓扑之间的中心位置以及它与高能物理的密切相互作用而变得重要。这一领域的主要问题是光滑单连通4-流形的分类。拓扑学和物理学之间的相互作用刺激了不变量的构造--首先是Donaldson的不变量，然后是Seiberg和Witten的不变量--它们有助于区分四维流形的微分同态类型。这些都带来了重大的进步，并使工人们能够研究4-流形的新结构。这些既混淆了分类问题，又活跃了理论，增强了理论的丰富性。这一理论现在甚至连猜测的分类都没有。似乎还需要更多的例子来确定合适的分类方案，而提出者打算在这样的结构上工作。与理论物理有非常密切联系的一类4-流形是那些具有辛结构的流形，在过去的几年里，辛4-流形理论也取得了进展，特别是新的构造，最值得注意的是，Donaldson关于Lefschetz纤维的工作。作者打算继续研究辛子流形上Lefschetz原函数的显式构造及相关问题。提出者的最终目标是开发光滑4流形的新结构，希望开始出现一个大致的图景。这个项目的重点将是构造光滑和辛四维流形的新类型的例子，并研究嵌入辛子流形(直到光滑同调)在给定的同调类中的多样性。特别地，复射影平面上的每个辛曲面是否都光滑地等同于一条全纯曲线？如果一个人允许足够多的爆破，这就不是真的。另一个问题是单连通可约4-流形的地理问题。每个这样的流形都可以在平面上被分配一个与其特征数相对应的格点。问题是要研究实现了哪些点。已经有了显著的进展，但仍有许多工作要做，主要研究者计划寻找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。此外，他和R·斯特恩一起猜想了4-辛流形的Noether不等式的一个替代品，他们有一个很有前途的证明方法，他们计划继续研究。

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.