Smooth and Symplectic 4-Manifolds

摘要：光滑4-流形理论由于其在低维和高维拓扑之间的中心位置以及与高能物理的密切相互作用而获得其重要性。该领域的主要问题是光滑单连通4流形的分类。拓扑和物理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——这些不变量对区分4流形的微分同构类型很有用。这些都导致了重大的进步，它们使工人们能够研究4流形的新结构。这些都给分类问题带来了困惑，但也给分类理论注入了活力，增强了分类理论的丰富性。这个理论现在甚至没有一个推测的分类。似乎还需要更多的例子来确定一个合适的分类方案，而提议者打算在这样的结构上工作。一类具有辛结构的4-流形与理论物理的关系异常密切，最近几年辛4-流形理论也取得了进展；尤其是新的结构，最值得注意的是唐纳森关于莱夫谢茨纤维的研究。作者打算继续研究Lefschetz颤振的显式构造和辛子流形的相关问题。提议者的最终目标是开发光滑流形的新结构，希望能够开始出现一个总体的图景。本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辛子流形的多样性（直到光滑同位素）。特别是，是否复射影平面上的每一个辛曲面都平滑地与全纯曲线同位素？如果一个人允许足够多的爆发，这是不正确的。另一个问题是单连通可约4流形的地理问题。每一个这样的流形都可以在平面上指定一个与它的特征数相对应的点阵点。问题是研究哪些点实现了。虽然已经取得了显著的进展，但仍有许多工作要做，主要研究者计划寻找新的方法来构造不可约的正签名单连通4流形。这些技术涉及到一种带约束的最小属曲面理论。此外，他和R. Stern一起，推测了辛4流形的诺特不等式的替代，他们有一个很有前途的技术来证明它，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.