Higher dimensional varieties and their applications

第一部分建立了代数簇的模的几何意义紧化，特别是对于一般类型的簇。一个关键的问题是理解规范层不是局部自由的变种的适当局部形变理论。其他部分集中于各种特殊类型的代数簇。从许多角度来看，有理连通簇是最简单的代数变体，但它们在有限域上的行为仍然很少被理解。作者的目的是在它们上找到定义在有限域上的有理曲线。作为他早期关于环在奇异环上作用的工作的推广，作者将研究如何通过对Orlik型猜想进行几何解释来计算环的积分同调.为了指定一个球面，人们只需要知道1个数：它的半径.对于椭球体，需要指定3个数字：3个半轴的长度。代数簇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。提出者的目的是从总体上研究这个问题。第一个问题是描述必要的参数，需要多少参数，以及这些参数与底层几何图形的关系。拟议研究的主要部分旨在了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，这些都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.