Arithmetic Groups in Algebraic Geometry

本课题的目标是研究复杂代数几何中出现的某些特殊模空间的几何和拓扑。这里，“特殊”意味着它们可以被离散群描述为厄米对称空间的商。（令人惊讶的是，有很多这样的例子。）第一个具体目标是了解这些空间的一大族的拓扑结构，其中包括极化光滑K3曲面的模空间和奇点理论中出现的其他几个例子。通过“理解的拓扑”，我们的意思是我们相信这些空间具有可收缩的全称覆盖，并且我们想要证明这一点。我们的想法是利用负弯曲度量空间理论中的工具来推广负弯曲黎曼流形的概念。第二个具体目标是与J. Carlson和D. Toledo共同完成一个项目，证明四维射影空间中三次超曲面的模空间与某离散群的复10球的商同构。第三个具体目标是关闭希尔伯特第14个问题的最后一个开放方面，通过展示二维可加群的一种表示，其不变量环不是有限生成的。该项目解决了代数几何中关于各种物体（如曲线和曲面）分类的具体问题。代数几何作为一门学科，研究可以用方程来定义的曲线和曲面，以及这些曲线和曲面的高维版本。举一个例子来说明这个分类问题，一个圆本质上和一个椭圆是一样的，因为它们中的一个可以通过在一个方向上拉伸平面而从另一个得到。椭圆本质上和抛物线是一样的，因为你可以想象椭圆的一端是固定的，而另一端是无穷大的。你可以进一步把这一端推到无穷远处，这样它就会出现在平面的另一边，你就会看到一个双曲线。一个代数几何学者通过说任意两个二次曲线是“投影等价的”来表达这些思想。这些形状的共同之处在于，它们都是由简单的方程定义的——两个变量，而且只有2次或更少的项。现代代数几何的一个主要部分是研究类似的问题，但随着变量数量的增加，或方程的复杂性增加，以允许更高次的多项式。例如，并非平面上所有由三次方程定义的曲线都是投影等价的。这些曲线彼此不同的方式对数学甚至物理的许多领域都非常重要。这个例子在19世纪就被理解了，但类似的问题仍然存在。当前提议的一个目的是获得对四维空间中三次“曲面”的射影等价类如何变化的理解。我们相信这可以用10维复空间中的单位球以一种美丽而惊人的方式来描述。这个结果与平面上三次曲线的经典结果相似。我们想要证明我们的直觉是正确的。另一个需要解决的问题是完成希尔伯特第14个问题的一部分，这是大卫·希尔伯特在20世纪初提出的著名问题之一。

The goal of the project is to study the geometry and topology of certain special moduli spaces arising in complex algebraic geometry. Here, "special" means that they may be described as quotients of Hermitian symmetric spaces by discrete groups. (There are surprisingly many examples of these spaces.) The first specific goal is to understand the topology of a large family of these spaces that includes the moduli spaces of polarized smooth K3 surfaces and several other examples arising in singularity theory. By "understand the topology of" we mean that we beleive these spaces have contractible universal covers, and we want to prove this. The idea is to use tools from the theory of negatively-curved metric spaces that generalize the idea of a negatively-curved Riemannian manifold. The second specific goal is to complete a joint project with J. Carlson and D. Toledo, to prove that the moduli space of cubic hypersurfaces in 4-dimensional projective space is isomorphic to a quotient of the complex 10-ball by a certain discrete group. The third specific goal is to close the last open aspect of Hilbert's 14th problem, by showing that there is a representation of the 2-dimensional additive group for which the ring of invariants is not finitely generated. The project addresses concrete problems in algebraic geometry concerning the classification of various objects like curves and surfaces. Algebraic geometry as a subject deals with curves and surfaces that can be defined by means of equations, and also with higher-dimensional versions of these curves and surfaces. An example giving the flavor of the classification problem is that a circle is essentially the same as an ellipse, because one of them can be got from the other by stretching the plane in one direction. An ellipse is also essentially the same as a parabola, because you can imagine keeping one end of the ellipse fixed and pushing the other off to infinity. You can go even further and push the end off past infinity, so that it reappears on the other side of the plane, and you see a hyperbola. An algebraic geometer expresses these ideas by saying that any two conics are "projectively equivalent". What these shapes all have in common is that they are defined by simple equations--two variables, and only terms of degree two or less. A major part of modern algebraic geometry is studying similar questions but with the number of variables increased, or the complexity of the equations increased to allow higher-degree polynomials. For example, it turns out that not all curves in the plane that are defined by degree three equations are projectively equivalent. The ways in which these curves can differ from each other is very important for many fields of mathematics and even physics. This example was understood in the 19th century, but similar problems remain open. One purpose of the current proposal is to gain an understanding of how the projective equivalence classes of the degree-three "surfaces" in four-dimensional space can vary. We believe that this may be described in a beautiful and surprising way using the unit sphere in 10-dimensional complex space. This result would be similar to a classical result for degree three curves in the plane. We want to prove that our intuition is right. Another problem to address is that of finishing a part of Hilbert's 14th problem, one of the celebrated problems posed by David Hilbert at the beginning of the 20th century.