Kaehler manifolds of constant curvature with conical singularities

常曲率度量包围着我们，我们生活在零曲率的欧几里得空间中，小肥皂泡有正的常曲率。具有恒定负曲率的物体不太常见，但它们确实以珊瑚和树叶的形状出现在自然界中。毫不奇怪，常曲率度量在几何拓扑学中起着重要的作用，它研究流形，即曲面的高维泛化。几何学家的梦想是在一个给定的流形上找到一个正则度规，这样它的拓扑，即它的形状直到拉伸和压缩，都将被它的几何所捕获。这个想法的一个著名的体现是瑟斯顿的几何化猜想解决了格里戈里佩雷尔曼。这个猜想给出了紧致3-流形允许常曲率度量的拓扑判据。这个项目的目标是研究常曲率流形的推广，即具有圆锥奇点的常曲率流形。这里，一个典型的例子是正四面体的表面（拓扑上是一个球体）。该表面在四面体的顶点处具有180度角的圆锥奇点，并且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坦表面。该项目的中心对象之一是高维概括的多面体表面，即多面体凯勒流形。高维多面体Kaehler流形与丰富的数学结构相连，并表现出很大的刚性，这可以通过下面的例子来说明。Hirzebruch指出，任何集合的3 n条线在复杂的投影平面与每一条线相交的其他n+1点，是一个集合的镜子复杂的反射组（一个复杂的类似物的结晶组）。事实证明，任何这样的线的集合是一个多面体凯勒度量在复平面上的奇异轨迹。这一结果为解决Hirzebruch猜想提供了一个合理的途径。寻找多面体Kaehler度量的存在对基础流形及其奇异轨迹的各种限制是这个项目的主要目标之一。回到曲面，我们注意到，具有圆锥奇点的平坦曲面是很好理解的。令人惊讶的是，这是不是在所有的情况下曲率一（即球面）表面与圆锥奇点。对这一问题的研究可以追溯到世纪初的费利克斯·克莱因的著作，但至今仍存在许多悬而未决的问题。例如，下面这个简单的问题直到2018年才得到解决：一个具有圆锥奇点的球面可以具有的圆锥角的所有可能集合是什么？这个问题的答案需要一些相关的工具，如抛物线束和胶合技术。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本项目的第二个主题。我们计划给出第一个完整的描述，这样的模空间的低维，我们将研究拓扑的高维模空间，并调查其自然映射到空间的黎曼曲面。值得注意的是，与球面的模空间相比，黎曼曲面的模空间的当前知识是非常广泛的，这个主题几乎与从可积系统到弦理论的所有几何学科有关。我们希望球面度量的模空间也能有类似的命运。

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.