Moduli Spaces, Hyperbolic Geometry, and Arithmetic Groups

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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.