Discrete Groups and Algebraic Geometry

该项目的主要问题是：（1）把真实的代数几何中的模空间作为真实的双曲轨道来研究，这是可能的，特别强调复双曲空间的Deligne-Mostow代数的真实的形式;（2）证明了具有给定度极化的K_3曲面的模空间具有可收缩的万有覆盖，（3）试图构造任意大维数的复双曲反射群，或证明它们不存在;（4）看怪物单群是否以某种简单的方式表示为复双曲13-空间的某种商。 这些项目的共同点是离散群在厄米特空间（如球和IV型域）上的作用在代数几何中的作用。 除了这些离散群之外，还有许多其他重要的群，如超平面补的基本群，以及这些基本群的某些子群（也许包括怪物）。对称性的研究被称为群论;群是图像、图案等的所有自变换的集合，这样就不会影响图片、图案等。 重点是变换，例如，围绕平面上的一点旋转图片的行为。 如果图像在旋转前后看起来完全相同，则它具有旋转对称性。 为了强调这一观点，我们说群体对图像“起作用”。 具有群作用的物体种类繁多，其中包括一些难以形象化但在物理学和数学中仍然非常重要的物体。 这些难以形象化的对象之一被称为复双曲空间，自19世纪世纪以来，许多数学家一直在研究群作用于复双曲空间的可能方式。 复双曲空间中最容易理解的变换被称为“复反射”，它（尽管如此）是一种绕轴的旋转。由这种变换产生的群在代数几何领域中起着特殊的作用，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“二元量子”和“K3曲面”。 研究人员还注意到一些目前无法解释的模式，这些模式可能将复杂的反射与一个名为“怪物”简单群的著名群联系起来。 该项目所要解决的具体技术问题都有助于我们进一步理解复反射产生的群，它们作用于复双曲空间。

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.