Higher dimensional varieties and their applications

该提议旨在研究代数变体的各个方面。第一部分旨在建立代数变量，特别是一般型变量的模的几何意义紧化。一个关键的问题是如何理解规范轴不是局部自由的变量的局部变形理论。其他部分着重于代数变体的各种特殊类别。从许多角度来看，合理连通的变分是最简单的代数变分，但是人们对它们在有限域上的行为仍然知之甚少。提出者的目标是在它们上面找到限定在有限域上的有理曲线。作为他早期关于奇点连杆上的圆作用的工作的延伸，提议者将研究如何通过对Orlik猜想的几何解释来计算连杆的积分同调。为了确定一个球体，我们只需要知道一个数字：它的半径。对于椭球体，需要指定3个数字：3个半轴的长度。代数变量的模理论旨在为可以用代数方程描述的更复杂的几何对象建立类似的模式。提议者的目的是一般地研究这个问题。第一个问题是描述必要的参数，需要多少个参数以及这些参数如何与底层几何结构相关联。所提出的研究的主要部分旨在理解当一个或多个参数变得非常大时发生的几何转变。对于许多应用来说，这些是与模理论相关的最有趣的现象。

The proposer aims to study various aspects of algebraic varieties.The first part intends to establish a geometrically meaningfulcompactification of the moduli of algebraic varieties, especially forvarieties of general type. A key problem is to understand the appropriatelocal deformation theory for varieties whose canonical sheaf is not locally free. The other parts focus on various special classes of algebraic varieties. Rationally connected varieties are the simplest algebraic varietiesfrom many points of view, but their behavior over finite fields is stillvery poorly understood. The proposer aims to find on them rational curvesdefined over finite fields. As an extension of his earlier work on circle actions on links of singularities, the proposer will study how to computethe integral homology of links by giving a geometric explanation to the conjectures of Orlik.In order to specify a sphere, one just needs to know 1 number: its radius. For an ellipsoid, one needs to specify 3 numbers: the lengths of the 3 semi axes. The theory of moduli of algebraic varieties aims to establish a similar pattern for more complicated geometric objects that can be described by algebraic equations. The proposer aims to study this question in general. The first question is to describe the necessary parameters, how many one needs and how these parameters relate to the underlying geometry. The main part of the proposed research aims to understand the geometric transitions that occur when one or more of the parameters become very large. For many applications, these are the most interesting phenomena related to moduli theory.