Higher Dimensional Algebraic Geometry

高维代数几何

• 批准号: 0554697
• 负责人: Sandor Kovacs
• 金额: $ 14.62万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554697&HistoricalAwards=false
• 关键词: Higher; Dimensional; Algebraic; Geometry

The investigator will work on several problems in higher dimensionalalgebraic geometry. In a joint project with Alexeev, Hassett, andKoll\'ar he plans to complete the proof of existence of coarsecomplete moduli spaces of stable log surfaces, an analog of the modulispace of stable curves. In another project the investigator is goingto work on boundedness, rigidity and hyperbolicity of subvarieties ofmoduli stacks of canonically polarized smooth projective varieties. These questions evolved from a landmark conjecture of Shafarevich, andits solution by Arakelov and Parshin, which played an important rolein Faltings' proof of the Mordell Conjecture. Part of this project isjoint work with Kebekus. In a third project the investigator andHacon are going to study the impact of the existence of nowherevanishing differential forms on the geometry of the underlyingvariety. Their goal is to prove several outstanding conjectures in thearea. In a fourth project the investigator and Araujo will worktoward proving a conjecture of Beauville giving a new characterizationof projective spaces and quadric hypersurfaces.This research is in the field of algebraic geometry, one of the oldestparts of modern mathematics, but one that blossomed to the point whereit has solved problems that have stood for centuries. Originally, andstill in its simplest form it treats figures defined in the plane bypolynomials. Today, the field uses methods not only from algebra, butalso from analysis and topology, and conversely it is extensively usedin those fields. Moreover it has proved itself useful in fields asdiverse as physics, theoretical computer science, cryptography, codingtheory and robotics. A central problem in algebraic geometry is theclassification of all geometric objects. In turn, an important part ofclassification theory is the theory of moduli. The latter's core ideais that one does not only want to understand these objects, but alsounderstand the way they can be deformed. Moduli spaces play a veryimportant role in theoretical physics. Studying curves on modulispaces provides information on how an object is changing inspace-time. One of the focuses of this project is on compact modulispaces. Those are extensions of moduli spaces in general and they giveadditional information about singular deformations, ones that areessentially different from others. Other goals of the project involvea better understanding of certain higher dimensional varieties.

研究者将致力于解决高维代数几何中的几个问题。在与 Alexeev、Hassett 和 Koll'ar 的联合项目中，他计划完成稳定对数曲面的粗完备模空间存在性的证明，这是稳定曲线模空间的模拟。 在另一个项目中，研究者将研究正则偏振光滑射影簇的模堆栈子簇的有界性、刚性和双曲性。这些问题源自沙法列维奇的一个里程碑式的猜想，以及阿拉克洛夫和帕尔辛的解答，在法尔廷斯证明莫德尔猜想中发挥了重要作用。该项目的一部分是与 Kebekus 的联合工作。 在第三个项目中，研究者和哈康将研究无处消失的微分形式的存在对基础品种几何形状的影响。他们的目标是证明该领域的几个杰出猜想。 在第四个项目中，研究者和阿劳霍将致力于证明博维尔猜想，为射影空间和二次超曲面提供新的表征。这项研究属于代数几何领域，它是现代数学最古老的部分之一，但却发展到了这一点。它解决了几个世纪以来存在的问题。最初，并且仍然以其最简单的形式，它处理由多项式在平面中定义的图形。 如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域中得到了广泛的应用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等各个领域都很有用。代数几何的一个中心问题是所有几何对象的分类。反过来，分类理论的一个重要组成部分是模理论。后者的核心思想是，人们不仅要了解这些物体，还要了解它们变形的方式。模空间在理论物理中起着非常重要的作用。 研究模空间上的曲线可以提供有关物体在时空中如何变化的信息。该项目的重点之一是紧凑模空间。这些是一般模空间的扩展，它们提供了有关奇异变形的附加信息，这些变形与其他变形有本质上的不同。该项目的其他目标包括更好地理解某些更高维度的品种。