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Moduli of Varieties of General Type

通用型品种模数

基本信息

批准号：
1901855
负责人：
Janos Kollar
金额：
$ 60万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901855&HistoricalAwards=false
关键词：
Moduli Varieties General Type

项目摘要

Many shapes that occur in nature are given by polynomial equations, and computers are especially fast in computing polynomials. Understanding the solutions of systems of polynomial equations is a key to advances in many areas, including computer aided geometric design, robotics and physics. In most cases a small change of the coefficients results in a small change of the geometric objects. For example, increasing the radius of a sphere a little results in a slightly larger sphere. The aim of this project is to give a general theoretical framework for analyzing those cases when a small change of the coefficients results in a dramatic change of the geometric objects. These are the cases when a small error in measurement or manufacturing may have large consequences.The main aim of the project is to understand what a good family of algebraic varieties is. The moduli of curves and its compactification constructed by Deligne and Mumford are among the most important objects in mathematics and in string theory. The goal of higher dimensional moduli theory is to construct more general versions and to use this in understanding the geometry of higher dimensional algebraic varieties. An approach to compactifying the moduli space of varieties of general type was proposed by Kollar and Shepherd-Barron in 1988. The limiting objects are called stable varieties. These satisfy a local condition (having only log-canonical singularities) and a global condition (having ample canonical class). More generally, following Alexeev, one should consider moduli spaces of pairs of a projective, reduced scheme and a non-negative linear combination (with rational or real coefficients) of divisors. In order to get a reasonable moduli space, these pairs should satisfy a local condition (the pair has only semi-log-canonical singularities) and a global condition (having ample log canonical class). The aim of the project is to apply the study of semi-log-canonical singularities to complete the moduli theory of stable pairs. The theory is complete as far as the underlying varieties are concerned, but the divisor parts of the pairs exhibit non-flat behavior that is harder to approach with the usual techniques. For stable varieties without divisors, the existence of the moduli space are known, but it needs to be written down in a systematic way. When a divisor is added, we run into the problem (first observed by Hassett) that a deformation of the pair need not induce a flat deformation of the divisor part. For seminormal base spaces the theory of Chow varieties is ideal to understand non-flat deformations over. The first main aim is to generalize this to reduced base spaces. In general we may expect a better theory when the coefficients in the divisorial part are bigger than a half. The second main aim is to work out the moduli theory by treating the variety and the divisor as essentially independent objects varying flatly. A third aim is to understand what happens when the coefficients in the divisorial part are allowed to be a half, since this case comes up frequently in applications. Besides being important in its own right, this is a test case for the various techniques that one can use in general. The fourth, most speculative part is to develop a general moduli theory that works optimally in all cases. A key problem is that there are several competing definitions that give slightly different answers for some non-reduced schemes. We need to understand the precise relationships and to find the optimal theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

自然界中出现的许多形状都是由多项式方程给出的，而计算机在计算多项式方面特别快。理解多项式方程组的解是许多领域进步的关键，包括计算机辅助几何设计、机器人和物理学。在大多数情况下，系数的微小变化会导致几何对象的微小变化。例如，稍微增加球体的半径会使球体稍微大一些。这个项目的目的是给出一个一般的理论框架，用于分析当系数的微小变化导致几何对象的剧烈变化时的情况。在这些情况下，测量或制造中的一个小错误可能会造成很大的后果。该项目的主要目的是了解什么是好的代数变种族。由Deligne和Mumford构造的曲线模及其紧化是数学和弦理论中最重要的研究对象之一。高维模理论的目标是构建更一般的版本，并使用它来理解高维代数变体的几何。Kollar和Shepherd-Barron在1988年提出了一种紧化一般类型变种模空间的方法。极限对象称为稳定变量。它们满足局部条件（仅具有对数规范奇点）和全局条件（具有充足的规范类）。更一般地说，遵循Alexeev，我们应该考虑一个投影、简化格式和除数的非负线性组合（具有有理或实系数）的对的模空间。为了得到合理的模空间，这些对应该满足一个局部条件（对只有半对数正则奇点）和一个全局条件（有足够的对数正则类）。本课题的目的是应用半对数正则奇点的研究来完成稳定对的模理论。就潜在的变量而言，这个理论是完整的，但是对的除数部分表现出非平坦的行为，用通常的技术很难接近。对于无除数的稳定变分，模空间的存在性是已知的，但需要系统地写下来。当添加除数时，我们遇到了一个问题（首先由Hassett观察到），即对的变形不必引起除数部分的平面变形。对于半正规基空间，Chow变分理论是理解非平坦变形的理想方法。第一个主要目标是将其推广到约简基空间。一般来说，当除数部分的系数大于1 / 2时，我们可以期望一个更好的理论。第二个主要目的是通过将变数和除数视为本质上独立的平变对象来推导模理论。第三个目标是理解当除数部分的系数被允许为1 / 2时会发生什么，因为这种情况在应用中经常出现。除了本身很重要之外，这也是一个测试用例，可以测试通常可以使用的各种技术。第四部分，也是最具思考性的部分，是建立一个在所有情况下都能最优工作的一般模理论。一个关键问题是，有几个相互竞争的定义，它们对一些非约简方案给出了略微不同的答案。我们需要了解它们之间的确切关系，并找到最优的理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。