Arithmetic Groups in Algebraic Geometry

该项目的目标是研究在复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特对称空间的子空间。 （令人惊讶的是，这些空间的例子很多。 第一个具体的目标是了解这些空间的一个大家族的拓扑结构，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有可收缩的泛覆盖，我们想证明这一点。 这个想法是使用负弯曲度量空间理论中的工具，推广了负弯曲黎曼流形的思想。 第二个具体目标是完成一个与J. Carlson和D. Toledo等，证明了四维射影空间中三次超曲面的模空间同构于复10-球的商。 第三个具体目标是关闭希尔伯特的第14个问题的最后一个开放方面，通过显示有一个表示的2维加法群的环的不变量是不确定生成的。该项目解决了代数几何中关于曲线和曲面等各种对象的分类的具体问题。 代数几何作为一门学科涉及可以通过方程定义的曲线和曲面，以及这些曲线和曲面的高维版本。 给出分类问题的一个例子是，圆与椭圆本质上是相同的，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。 椭圆本质上也和抛物线一样，因为你可以想象保持椭圆的一端固定，而将另一端推向无穷远。 你可以更进一步，把端点推到无穷远，这样它就重新出现在平面的另一边，你就看到了一条双曲线。 一个代数几何表达这些想法说，任何两个二次曲线是“射影等价”。 这些形状的共同点是它们都是由简单的方程定义的--两个变量，只有二次或更少的项。现代代数几何的一个主要部分是研究类似的问题，但变量的数量增加了，或者方程的复杂性增加了，允许更高次的多项式。 例如，事实证明，并非所有由三次方程定义的平面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学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。 目前的建议的一个目的是获得一个了解如何在四维空间的度三“表面”的投影等价类可以改变。 我们相信，这可以用10维复空间中的单位球以一种美丽而令人惊讶的方式来描述。 这个结果将类似于平面上三次曲线的经典结果。 我们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问题的一部分，这是大卫希尔伯特在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.