Moduli Spaces, Hyperbolic Geometry, and Arithmetic Groups

本项目研究三次超曲面模空间的几何结构。 其主要目的是发展三次三重模空间的复双曲几何和三次四重模空间的局部对称几何。 它也将开发相应的真实的三次超曲面的模空间的几何，它是由三重的情况下的真实的双曲片组装，并在四重的情况下的真实的双曲片的产品。 这种几何学产生于适当的周期映射，它的完整理解涉及到对奇点的相当详细的研究和对判别式的更深入的理解。 因此，它预计将带来新的见解奇点理论，以及到理论的离散群。 对于三次曲线，模空间几何的研究一直是数学的中心领域，有许多应用和分支。 对于三次曲面，这种几何学最近由研究者和Allcock在复域和真实的域中发展起来。 应用这种几何算术问题和表示离散群作用于非积极弯曲空间看起来非常有前途，将在这个项目中开发。这个一般的研究领域涉及被称为模空间的空间的研究。 其中心思想是，几何对象的集合本身就是一个新的实体，具有自己的几何形状。 这个新的实体被称为原始对象的模空间。 原始物体的几何形状和模空间的几何形状之间存在着丰富的相互作用，例如，一个物体的对称性在另一个物体的对称性中反映出来。 最简单的例子是平面上三角形的形状：每个形状都由边的比率决定，模空间是这些比率的集合，平面上的三角形具有自己的几何形状。 这个例子本质上描述了一个非常经典和核心的对象，三次曲线的模空间，历史上，它一直是最重要的模空间。 其关系是三次曲线是拓扑环面，环面是由平面上的两个三角形构成的。 本项目的主旨是发展由三次方程定义的高维物体的模量几何。 研究三次方程的原因是，它是最低的程度，他们的解决方案的形状可以连续变化。 已知它们的模空间在某些情况下具有非常特殊的几何（技术上称为局部对称，在某些情况下称为双曲）。 似乎这应该是真实的通过第四维度。 三次方程的解的模的这种特殊几何的存在的原因在高维中不像在三次曲线的情况下那样明显。 这使得他们的研究更加困难，但从长远来看，它有望有许多有趣的应用。

This project studies geometric structures on moduli spaces of cubic hypersurfaces. Its principal aim is to develop the complex hyperbolic geometry of the moduli space of cubic threefolds, and the locally symmetric geometry of the moduli space of cubic fourfolds. It will also develop the geometry of the moduli space of the corresponding real cubic hypersurfaces, which is assembled from real hyperbolic pieces in the case of threefolds, and of products of real hyperbolic pieces in the case of fourfolds. This geometry arises from suitable period maps, and its complete understanding involves a quite detailed study of singularities and a deeper understanding of the discriminant. It is thus expected to bring new insights into singularity theory, as well as into the theory of discrete groups. For the case of cubic curves, the study of the geometry of the moduli space has been a central area of mathematics, with many applications and ramifications. For cubic surfaces this geometry has been developed recently, both in the complex and the real domains, by the investigators and Allcock. Applications of this geometry to arithmetic questions and to representations of discrete groups acting on non-positively curved spaces look very promising and will be developed in this project. This general area of research concerns the study of spaces known as moduli spaces. The central idea is that a collection of geometric objects is itself a new entity with its own geometry. This new entity is called the moduli space of the original objects. There is a rich interplay between the geometry of the original objects and the geometry of the moduli space, for example the symmetries of one reflect themselves in the symmetries of the other. The simplest example would be the shapes of triangles in the plane: each shape is determined by the ratios of the sides, the moduli space is the collection of these ratios, a triangle in the plane with its own geometry. This example in essence describes a very classical and central object, the moduli space of cubic curves, which, historically, has been the most important moduli space. The relation is that cubic curves are topologically tori, and tori are built by doubling triangles in the plane. The thrust of the present project is to develop the geometry of moduli of higher dimensional objects defined by cubic equations. The reason for studying cubic equations is that it is the lowest degree where the shapes of their solutions can vary continuously. Their moduli spaces are known in some cases to have a very special geometry (technically called locally symmetric, in some cases hyperbolic). It seems that this should be true through dimension four. The reasons for the presence of this special geometry of the moduli of solutions of cubic equations are not as transparent in higher dimensions as they are in the case of cubic curves. This makes their study more difficult, but in the long term it is expected to have many interesting applications.