Floer homology for immersed Lagrangian submanifolds

大多数现代几何研究某种空间。在微分几何中考虑的空间被称为流形，局部看起来像n维欧几里得空间，但整体上有一个有趣的形状。如果流形是封闭的，没有边，那么流形就是紧的。甜甜圈的表面是一个紧致的二维流形。流形M的子流形N是M的子集，M本身也是流形，通常比M的维数小。有两种类型：嵌入子流形，它可能不会与自己相交（交叉）；浸入子流形，它可能与自己相交（交叉）。人们通常认为流形有一些额外的几何结构，比如黎曼度规，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。数学家喜欢它们，因为它们是为数不多的具有无限维局部对称的结构之一，这给辛几何带来了一种不同寻常的、完全全局的味道。拉格朗日子流形是辛流形的一种特殊的子流形。给定辛流形M的两个紧的、内嵌的拉格朗日子流形N，N*，在一定条件下可以定义花同调群HF(N,N*)，它们大致是有限维向量空间。这个定义是非常困难的。为此，在M上选择一个辅助复结构J，并在M中计算边界（边）为N与N*并的J个全纯二维圆盘D。关于HF（N,N*）值得注意的是它与J的选择无关，并且在拉格朗日子流形中移动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、重要的信息，这些信息是用其他已知的方法无法得到的。它是辛几何中的一个强大工具。本研究的主要目的是将花同调群HF（N,N*）的定义推广到浸入式拉格朗日子流形N，N*，并了解它们可以被定义的条件（定义的阻碍）。这涉及到的新技术问题与j全纯盘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*）的正确代数建立。我们还想了解不改变HF（N,N*）的浸入式拉格朗日子流形中N和N*的允许运动。除了辛几何学者感兴趣外，我们相信这些结果将对一些关于特殊拉格朗日几何和卡拉比-丘流形的主要猜想有重要的应用，这些猜想是弦理论物理学家感兴趣的。关键是，这些猜想只有在正确的拉格朗日算子类中才能成立，而嵌入的非奇异拉格朗日算子不是一个足够大的类。有充分的证据表明，合适的类别可能是浸入式拉格朗日算子，它的花同调是畅通的，但要理解这意味着什么，我们首先需要一个浸入式拉格朗日算子的花同调理论，这是我们希望发展的。

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.