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Kaehler manifolds of constant curvature with conical singularities

具有圆锥奇点的常曲率凯勒流形

基本信息

批准号：
EP/S035788/1
负责人：
Dmitri Panov
金额：
$ 40.41万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS035788%2F1
关键词：
Kaehler manifolds constant curvature conical

项目摘要

Constant curvature metrics surround us, we live in Euclidean space of zero curvature, little soap bubbles have positive constant curvature. Objects of constant negative curvature are less familiar, but they do appear in Nature in the shape of corals and leaves. Not surprisingly, constant curvature metrics play an important role in geometric topology, which studies manifolds, i.e. higher dimensional generalisations of surfaces. It is a geometer's dream to find a canonical metric on a given manifold so that its topology, i.e. its shape up to stretching and squeezing, will be captured by its geometry. One famous incarnation of this idea is Thurston's geometrization conjecture solved by Grigori Perelman. This conjecture gives topological criteria for a compact 3-manifold to admit a constant curvature metric. The goal of this project is to study a generalisation of constant curvature manifolds, namely constant curvature manifolds with conical singularities. Here, a prototypical example is the surface of a regular tetrahedron (which is topologically a sphere). This surface has conical singularities of angle 180 degrees at the vertices of the tetrahedron and is flat elsewhere. More generally, surfaces of all polyhedra are flat surfaces with conical singularities. One of the central objects of this project consists of higher-dimensional generalisations of polyhedral surfaces, namely polyhedral Kaehler manifolds. Higher-dimensional polyhedral Kaehler manifolds are connected to rich mathematical structures and exhibit a lot of rigidity, this can be illustrated by the following example. Hirzebruch conjectured that any collection of 3n lines in the complex projective plane with each line intersecting others in n+1 points, is a collection of mirrors of a complex reflection group (a complex analogue of a crystallographic group). It turns out that any such collection of lines is the singular locus of a polyhedral Kaehler metric on the complex plane. This result gives a plausible approach for settling the Hirzebruch conjecture. Looking for various restrictions that the existence of a polyhedral Kaehler metric imposes on the underlying manifold and its singular locus is one of the main goals of this project.Coming back to surfaces, we note that flat surfaces with conical singularities are quite well understood. Surprisingly, this is not at all the case for curvature one (i.e. spherical) surfaces with conical singularities. The study of this topic can be traced back to the beginning of 20th century and the work of Felix Klein, however it is full of open questions. For example, the following simple question was settled only in 2018.Question: what are all possible collections of conical angles that a spherical surface with conical singularities can have? The answer to this question required a number of involved tools, such as parabolic bundles and gluing techniques. An important feature of spherical surfaces with conical singularities is that the spaces of such metrics are interesting geometric objects in their own right. Investigation of such moduli spaces is a second theme of this project. We plan to give a first full description of such moduli spaces of low dimensions, we will study the topology of higher-dimensional moduli spaces and investigate their natural maps to the space of Riemann surfaces. It is worth noting that in contrast to moduli spaces of spherical surfaces, the current knowledge of moduli spaces of Riemann surfaces is extremely vast and this topic is connected to virtually all geometric disciplines from integrable systems to string theory. We hope that the moduli spaces of spherical metrics could have a similar fate.

常曲率度量围绕着我们，我们生活在零曲率的欧几里德空间中，小的肥皂泡具有正的常曲率。恒定负曲率的物体不太常见，但它们在自然界中确实以珊瑚和树叶的形状出现。不出所料，常曲率度量在几何拓扑学中扮演着重要的角色，几何拓扑学研究流形，即曲面的高维推广。几何学家的梦想是在给定的流形上找到一个正则度量，以便它的拓扑结构，即它的拉伸和挤压形状，将被它的几何所捕获。这一思想的一个著名化身是格里戈里·佩雷尔曼解决了瑟斯顿的几何化猜想。这一猜想给出了紧致3-流形接纳常曲率度量的拓扑准则。这个项目的目的是研究常曲率流形的一个推广，即具有锥形奇点的常曲率流形。这里，一个典型的例子是正四面体的表面(它在拓扑上是一个球体)。该曲面在四面体的顶点处具有180度角的圆锥奇点，而在其他地方是平坦的。更一般地，所有多面体的表面都是具有圆锥奇点的平面。这个项目的中心目标之一是多面体曲面的高维推广，即多面体Kaehler流形。高维多面体Kaehler流形与丰富的数学结构相联系，并表现出许多刚性，这可以通过下面的例子来说明。Hirzebruch猜想，复射影平面上3n条直线的任何集合，其中每条直线在n+1个点上与其他直线相交，就是一个复反射群(晶体群的复类比)的镜面的集合。证明了任何这样的直线集合都是复平面上多面体Kaehler度规的奇异轨迹。这一结果为解决Hirzebruch猜想提供了一条可信的途径。寻找多面体Kaehler度量的存在对基础流形及其奇异轨迹施加的各种限制是本项目的主要目标之一。回到曲面，我们注意到具有圆锥奇点的平面是相当好的理解。令人惊讶的是，对于具有圆锥奇点的曲率一(即球面)曲面来说，情况并非如此。对这一主题的研究可以追溯到20世纪初的费利克斯·克莱因的作品，但也充满了悬而未决的问题。例如，下面这个简单的问题直到2018年才得到解决。问题：具有圆锥奇点的球面可能有哪些圆锥角的集合？这个问题的答案需要许多复杂的工具，如抛物线捆绑和粘合技术。具有圆锥奇点的球面的一个重要特征是，这种度量的空间本身就是有趣的几何对象。对这种模空间的研究是这个项目的第二个主题。我们计划首先给出这种低维模空间的完整描述，我们将研究高维模空间的拓扑，并研究它们到Riemann曲面空间的自然映射。值得注意的是，与球面的模空间相比，目前关于黎曼曲面的模空间的知识非常广泛，这一主题几乎涉及从可积系统到弦理论的所有几何学科。我们希望球面度量的模空间也能有类似的命运。