Geometry of Moduli Spaces and Topology of Varieties

本课题研究模空间上的几何结构。最近研究者和Allcock关于稳定立方曲面的模空间是复双曲的定理表明，模空间上的复双曲结构比以前预期的更为普遍。该项目将发展稳定的del Pezzo曲面和稳定的三次立方的模空间的复杂双曲几何。我们将研究这种几何结构对这些空间拓扑结构的影响。我们期望光滑del Pezzo曲面和光滑立方三折的模空间是Eilenberg-MacLane空间，并且它们的基本群不是线性群，甚至可能不是剩余有限群。这些空间的实点的实双曲几何，以及与del Pezzo曲面和三次三折的实代数几何的关系，也将得到发展。与辛几何和自同构形式的联系也被期待。一般的研究领域涉及被称为模空间的空间的研究。中心思想是几何物体的集合有其自身的几何形状。这种新的几何形状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存在着丰富的相互作用，例如，一个物体的对称性会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：每个形状都是由边的比率决定的，模空间是这些比率的集合，一个平面上的三角形有自己的几何形状。这个简单的例子很快引出了非常复杂的例子：我们可以采用更复杂的多边形或多面体。或者我们可以将平面上的格子形状的研究简化为这个例子（比如平坦的环面，椭圆曲线，复数上的三次曲线）。代数几何给出了许多有趣的模空间的例子，其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.