Discrete Groups and Algebraic Geometry

本课题研究的问题是：(1)将实代数几何中的模空间作为可能的实双曲轨道进行研究，重点研究复双曲空间的delign - mostow商的实形式；(2)证明了具有给定极化度的K3曲面的模空间具有可收缩的普适覆盖，并证明了与任意Coxeter群相关的复超平面排列的补具有类似的结果；(3)尝试构造任意大维的复杂双曲反射群，或证明它们不存在；(4)在复双曲13空间的某一商数上，看怪物单群是否以某种简单的方式表现出来。这些项目的共同主线是作用于厄米对称空间（如球和IV型域）的离散群在代数几何中的作用。除了这些离散群之外，还有一些其他重要的群，比如超平面补的基本群，以及这些基本群的某些商（可能包括怪物）。对对称性的研究被称为群论；群是一幅图片、图案等的所有自我变换的集合，这些变换不影响图片、图案等。重点是变换，例如，围绕平面上的一点旋转图像的行为。如果图像在旋转前后看起来完全一样，那么它就具有旋转对称性。为了强调这个观点，我们说这个群体在画面上“行动”。有许多不同种类的物体具有群体作用，包括一些难以想象但在物理和数学中仍然非常重要的物体。其中一种难以想象的物体被称为复双曲空间，自19世纪以来，许多数学家一直在研究群在复双曲空间上活动的可能方式。最容易理解的复双曲空间的变换被称为“复反射”，这是一种围绕轴的旋转（尽管名字如此）。由这种变换产生的群在代数几何领域中发挥着特殊的作用，有助于解释在研究一些重要对象时出现的某些模式，包括所谓的“二元量化”和“K3曲面”。研究者还注意到一些目前无法解释的模式，这些模式可能将复杂反射与一个著名的被称为“怪物”的简单群联系起来。该项目所要解决的具体技术问题都试图提高我们对复杂反射产生的群体的理解，这些群体作用于复杂的双曲空间。

The problems comprising the project are (1) to study moduli spaces inreal algebraic geometry as real hyperbolic orbifolds, which this ispossible, with particular emphasis on the real forms of theDeligne-Mostow quotients of the complex hyperbolic space; (2) to provethat the moduli space of K3 surfaces with a polarization of given degreehas contractible universal cover, and a similar result for thecomplement of the complex hyperplane arrangement associated to an arbitrary Coxeter group; (3) to try to construct complex hyperbolicreflection groups in arbitrarily large dimensions, or prove that they do not exist; and (4) to see if the monster simple group manifests itself in a certain simple way in terms of a certain quotient of complex hyperbolic 13-space. The common thread in these projects is the role inalgebraic geometry of discrete groups acting on Hermitian symmetricspaces like the ball and the type IV domains. Besides these discretegroups, there are a number of other important groups involved, like thefundamental groups of hyperplane complements, and certain quotients ofthese fundamental groups (perhaps including the monster).The study of symmetry is called group theory; a group is the collectionof all self-transformations of a picture, pattern, etc., that leave the picture, pattern, etc. alone. The focus is on the transformation, for example, the act of rotating a picture around apoint in the plane. If the picture looks exactly the same before and after the rotation, then it has rotational symmetry. To emphasize this point of view we say that the group "acts" on the picture. There are lots of different sorts of objects with group actions, including somethat are difficult of visualize but are still very important in physics and mathematics. One of these difficult-to-visualize objects is calledcomplex hyperbolic space, and the possible ways that groups can act oncomplex hyperbolic space have been studied by many mathematicians sincethe 19th century. The easiest-to-understand transformations of complexhyperbolic space are called "complex reflections", which (despite thename) are a sort of rotation around an axis. Groups that are generatedby this kind of transformation play a privileged role in the field ofalgebraic geometry, helping to explain certain patterns which appearwhen studying some important objects, including what are called "binaryquantics" and "K3 surfaces". The investigator has also noticed somemore patterns, currently unexplained, which may connect complexreflections to a famous group called the "monster" simple group. Thespecific technical problems to be addressed by the project all attemptto advance our understanding of groups generated by complex reflections,acting on complex hyperbolic space.