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Floer homology for immersed Lagrangian submanifolds

浸入式拉格朗日子流形的 Florer 同调

基本信息

批准号：
EP/D07763X/1
负责人：
Dominic Joyce
金额：
$ 6.69万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2006
资助国家：
英国
起止时间：
2006 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD07763X%2F1
关键词：
Floer homology immersed Lagrangian submanifolds

项目摘要

Most of modern geometry studies some kind of space. The spaces considered in differential geometry are called manifolds , spaces which locally look like n-dimensional Euclidean space but globally have an interesting shape. A manifold is compact if it is closed up, with no edges. The surface of a doughnut is a compact 2-dimensional manifold. A submanifold N of a manifold M is a subset of M which is itself a manifold, usually of smaller dimension than M. There are two kinds: embedded submanifolds, which may not intersect (cross) themselves, and immersed submanifolds, which may. One usually considers manifolds with some extra geometric structure, such as a Riemannian metric , which tells you the lengths of paths in the manifold, or a symplectic structure , which tells you the areas of 2-dimensional submanifolds of the manifold. Symplectic manifolds are the foundation of the mathematical formulation of mechanics, and so of much of classical physics. They are also very interesting in their own right. Mathematicians like them because they are one of very few structures with an infinite-dimensional amount of local symmetry, which gives symplectic geometry an unusual, entirely global flavour. Lagrangian submanifolds are a special kind of submanifold of a symplectic manifold. Given two compact, embedded Lagrangian submanifolds N, N* of a symplectic manifold M, one can under certain conditions define the Floer homology groups HF(N,N*), which are roughly speaking finite-dimensional vector spaces. The definition is very difficult. To do it, one chooses an auxiliary complex structure J on M and counts J-holomorphic 2-dimensional discs D in M with boundary (edge) in the union of N and N*. The remarkable thing about HF(N,N*) is that it is independent of the choice of J, and is also unchanged by moving N and N* around amongst Lagrangian submanifolds. It encodes some mysterious, nontrivial information about Lagrangian submanifolds one cannot get at in any other known way. It is a powerful tool in symplectic geometry. The main aim of the research is to extend the definition of Floer homology groups HF(N,N*) to immersed Lagrangian submanifolds N, N*, and to understand the conditions under which they can be defined ( obstructions to their definition). The new technical problems this involves have to do with J-holomorphic discs D whose boundary passes through self-intersection points in N or N*, and what is the right algebraic set-up for including and counting these to get well-behaved groups HF(N,N*). We also want to understand the allowed motions of N and N* amongst immersed Lagrangian submanifolds which do not change HF(N,N*). As well as being interesting to symplectic geometers, we believe these results will have important applications to several major conjectures about special Lagrangian geometry and Calabi-Yau manifolds, which are of interest to physicists working in String Theory. The point is that these conjectures can only be true if one works in the right class of Lagrangians, and embedded nonsingular Lagrangians are not a large enough class. There is good evidence that the right class to consider may be immersed Lagrangians whose Floer homology is unobstructed, but to understand what this means we first need a theory of Floer homology for immersed Lagrangians, which we hope to develop.

大多数现代几何学研究某种空间。微分几何中考虑的空间称为流形，这些空间局部看起来像n维欧几里得空间，但整体上具有有趣的形状。一个流形是紧致的，如果它是封闭的，没有边。甜甜圈的表面是一个紧凑的二维流形。一个流形M的一个子流形N是M的一个子集，它本身也是一个流形，通常维数比M小。有两种类型：嵌入子流形，可能不相交（交叉）自己，浸入子流形，可能。人们通常考虑具有一些额外几何结构的流形，例如黎曼度量，它告诉你流形中路径的长度，或者辛结构，它告诉你流形的二维子流形的面积。辛流形是力学数学公式的基础，也是许多经典物理学的基础。它们本身也很有趣。数学家们喜欢它们，因为它们是极少数具有无限维局部对称性的结构之一，这给辛几何带来了一种不寻常的、完全全局的味道。拉格朗日子流形是辛流形的一种特殊子流形。给定辛流形M的两个紧致嵌入拉格朗日子流形N，N*，在一定条件下可以定义Floer同调群HF（N，N*），它们大致上是有限维向量空间。这个定义很难。为了做到这一点，我们在M上选择一个辅助复结构J，并计数M中的J-全纯二维圆盘D，其边界（边）在N和N* 的并中。HF（N，N*）的显著之处在于它与J的选择无关，并且在拉格朗日子流形之间移动N和N* 也不会改变。它编码了一些关于拉格朗日子流形的神秘的、非平凡的信息，人们无法以任何其他已知的方式获得这些信息。它是辛几何中的一个强有力的工具。研究的主要目的是将Floer同调群HF（N，N*）的定义扩展到浸入拉格朗日子流形N，N*，并了解它们可以被定义的条件（它们定义的障碍）。这涉及到的新技术问题与J-全纯圆D有关，其边界通过N或N* 中的自交点，以及包含和计数这些点以得到行为良好的群HF（N，N*）的正确代数设置是什么。我们还想了解N和N* 在不改变HF（N，N*）的浸入拉格朗日子流形之间的允许运动。以及有趣的辛几何学家，我们相信这些结果将有重要的应用，几个主要的approachures关于特殊的拉格朗日几何和Calabi-Yau流形，这是有兴趣的物理学家工作在弦理论。关键是，只有在正确的拉格朗日类中工作时，这些假设才能为真，而嵌入的非奇异拉格朗日不是一个足够大的类。有很好的证据表明，正确的类考虑可能是沉浸拉格朗日的弗洛尔同调是通畅的，但要理解这意味着什么，我们首先需要一个理论的弗洛尔同调沉浸拉格朗日，我们希望发展。