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Applications of homotopy theory to algebraic geometry

同伦理论在代数几何中的应用

基本信息

批准号：
EP/I004130/1
负责人：
Jonathan Pridham
金额：
$ 80.33万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI004130%2F1
关键词：
Applications homotopy theory algebraic geometry

项目摘要

In homotopy theory, topological spaces (i.e. shapes) are regarded as being the same if we can deform continuously from one to the other. Algebraic varieties are spaces defined by polynomial equations, often over the complex numbers; studying their homotopy theory means trying to tell which topological spaces can be deformed continuously to get algebraic varieties, or when a continuous map between algebraic varieties can be continuously deformed to a map defined by polynomials.If the polynomials defining a variety are rational numbers (i.e. fractions), this automatically gives the complex variety a group of symmetries, called the Galois group. Although these symmetries are not continuous (i.e. nearby points can be sent far apart), they preserve something called the etale topology. This is an abstract concept which looks somewhat unnatural, butbehaves well enough to preserve many of the topological features of the variety. Part of my project will involve investigating how the Galois group interacts with the etale topology. I also study algebraic varieties in finite and mixed characteristics. Finite characteristics are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. For instance, in characteristic 3 the equation 1+1+1=0 holds. In mixed characteristic, p need not be 0, but the sequence 1,p, p^2, p^3 ... converges to 0.Although classical geometry of varieties does not make sense in finite and mixed characteristics, the etale topology provides a suitable alternative, allowing us to gain much valuable insight into the behaviour of the Galois group. This is an area which I find fascinating, as much topological intuition still works in contexts far removed from real and complex geometry. Indeed, many results in complex geometry have been motivated by phenomena observed in finite characteristic.Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to those of algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry. To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can themselves have symmetries and so on, giving rise to infinity stacks.Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities (points where the space is not smooth), and regions of different dimensions. Partially inspired by ideas from theoretical physics, it has been conjectured that every moduli stack can be extended to a derived moduli stack, which would have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness . Different classification problems can give rise to the same moduli stack, but different derived moduli stacks. Much of my work will be to try to construct derived moduli stacks for a large class of problems. This has important applications in algebraic geometry, as there are many problems for which the moduli stacks are unmanageable, but which should become accessible using derived moduli stacks. I will also seek to investigate the geometry and behaviour of derived stacks themselves.A common thread through the various aspects of my project will be to find ways of applying powerful ideas and techniques from a branch of topology, namely homotopy theory, in contexts where they would not, at first sight, appear to be relevant.

在同伦理论中，如果拓扑空间（即形状）可以从一个空间连续变形到另一个空间，则认为拓扑空间（即形状）是相同的。代数簇是由多项式方程定义的空间，通常是复数；研究它们的同伦理论意味着试图找出哪些拓扑空间可以连续变形以获得代数簇，或者代数簇之间的连续映射何时可以连续变形为由多项式定义的映射。如果定义簇的多项式是有理数（即分数），这会自动赋予复簇一组对称性，称为伽罗瓦群。尽管这些对称性不是连续的（即附近的点可以发送到相距很远的地方），但它们保留了称为 etale 拓扑的东西。这是一个抽象的概念，看起来有些不自然，但表现得足够好，可以保留该品种的许多拓扑特征。我的项目的一部分将涉及研究伽罗瓦群如何与 etale 拓扑交互。我还研究有限和混合特征的代数簇。有限特征是通过选择素数 p 并将其设置为零来修改算术规则的宇宙。例如，在特性3中，等式1+1+1=0成立。在混合特征中，p 不必为 0，但序列 1,p, p^2, p^3 ... 收敛于 0。虽然经典的簇几何在有限和混合特征中没有意义，但 etale 拓扑提供了一个合适的替代方案，使我们能够对伽罗瓦群的行为获得许多有价值的见解。这是一个我觉得很有趣的领域，因为许多拓扑直觉在远离真实和复杂几何的环境中仍然有效。事实上，复杂几何中的许多结果都是由在有限特征中观察到的现象所激发的。模空间参数化几何对象的类，并且本身通常可以给出几何结构，类似于代数簇的几何结构。这种结构在参数化具有大量对称性的对象时往往会出现错误。为了避免这个困难，代数几何学家使用模堆栈，对对称性和对象进行参数化。有时，对称性本身可以具有对称性等，从而产生无限堆栈。通常，可以通过简单地计算定义其参数化的几何对象时的自由度来计算模堆栈的维数。然而，空间通常包含奇点（空间不平滑的点）和不同维度的区域。部分受到理论物理学思想的启发，人们推测每个模堆栈都可以扩展到派生模堆栈，该派生模堆栈将具有预期的维度，但某些维度只是虚拟的。扩展到这些虚拟维度还可以消除奇点，这种现象称为隐藏平滑。不同的分类问题可以产生相同的模堆栈，但导出的模堆栈不同。我的大部分工作将是尝试为一大类问题构建派生模堆栈。这在代数几何中具有重要的应用，因为有许多问题是模堆栈无法管理的，但应该可以使用派生模堆栈来解决。我还将寻求研究派生堆栈本身的几何形状和行为。贯穿我的项目各个方面的一个共同线索将是找到在乍一看似乎不相关的背景下应用来自拓扑学分支（即同伦理论）的强大思想和技术的方法。