Geometry of Moduli Spaces and Topology of Varieties

本计画研究模空间的几何结构。 最近的定理的调查和Allcock的模空间的stablecubic曲面是复杂的双曲线表明，复杂的双曲线结构上的模空间比以前预期的更常见。 该项目将开发复杂的双曲几何的模空间的稳定德尔佩佐表面和稳定的立方三倍。 将研究这种几何对这些空间拓扑的影响。 光滑del Pezzo曲面和光滑三重曲面的模空间是Eilenberg-MacLane空间，它们的基本群不是线性群，甚至可能不是剩余有限群。 这些空间的真实的点的实双曲几何，以及与del Pezzo曲面和三次三重曲面的真实的代数几何的关系，也将得到发展。 与辛几何和自同构形式的联系也是预期的。一般的研究领域涉及到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有其自身的几何结构. 这个新的几何体被称为原始物体的模空间。 原始物体的几何形状和模空间的几何形状之间有着丰富的相互作用，例如，一个物体的对称性反映在另一个物体的对称性中。 最简单的例子是平面上三角形的形状：每个形状由边长比决定，模空间是这些比值的集合，平面上的三角形有自己的几何形状。 这个简单的例子很快就引出了非常复杂的例子：我们可以用更复杂的多边形或多面体。 或者，我们可以把平面上格子形状的研究（比如平面环面，椭圆曲线，复数上的椭圆曲线）简化为这个例子。 代数几何给出了许多有趣的例子模空间的几何是非常活跃的研究。

This project studies geometric structures on moduli spaces. The recenttheorem of the investigators and Allcock that the moduli space of stablecubic surfaces is complex hyperbolic suggests that complex hyperbolicstructures on moduli spaces are more common than previously expected. Theproject will develop the complex hyperbolic geometry of the moduli spaces ofstable del Pezzo surfaces and of stable cubic threefolds. The consequencesof this geometry to the topology of these spaces will be studied. It isexpected that the moduli spaces of smooth del Pezzo surfaces and smooth cubicthreefolds are Eilenberg-MacLane spaces, and that their fundamental groupsare not linear groups, perhaps not even residually finite groups. The realhyperbolic geometry of the real points of these spaces, and the relation tothe real algebraic geometry of del Pezzo surfaces and cubic threefolds, willalso be developed. Connections with symplectic geometry and automorphicforms are also expected.The general area of research concerns the study of spaces known as modulispaces. The central idea is that a collection of geometric objects hasitself its own geometry. This new geometry is called the moduli space of theoriginal objects. There is a rich interplay between the geometry of theoriginal objects and the geometry of the moduli space, for example thesymmetries of one reflect themselves in the symmetries of the other. Thesimplest example would be the shapes of triangles in the plane: each shape isdetermined by the ratios of the sides, the moduli space is the collection ofthese ratios, a triangle in the plane with its own geometry. This simpleexample leads quickly to very intricate ones: one could take morecomplicated polygons or polyhedra. Or one could reduce the study of shapesof lattices in the plane (thus flat tori, thus elliptic curves, thus cubiccurves over the complex numbers) to this example. Algebraic geometry givesmany interesting examples of moduli spaces whose geometry is very activelystudied.