Birational Geometry and Hodge Theory

双有理几何和霍奇理论

• 批准号: 0500659
• 负责人: Donu Arapura
• 金额: --
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500659&HistoricalAwards=false
• 关键词: Birational; Geometry; Hodge; Theory

The project will be broken into various subprojects in algebraicgeometry. In the first subproject, Arapura intends to study vanishingtheorems, which are very roughly a collection of techniques forcontrolling homological invariants. In the second subproject, Arapuraintends to study some problems related to the Hodge conjecture, whichpredicts that certain homological entities called Hodge classes can berealized in a geometric way. In the fifth subproject, Matsuki plans tostudy the homological aspects of the minimal model program, and alsoto study the automorphisms of the three dimensional space. Severalsubproject involve finding good models, or resolutions, for varietiesand maps between them. For maps this can be formulated more preciselyas the toroidalization problem, and this will be studied by Matsuki inthe fourth subproject. Finding good models for a variety amounts toresolution of singularities. Various aspects of the problem will bestudied by Matsuki and Wlodarczyk in the seventh subproject. Inparticular, Wlodarczyk has found a simplified algorithm forresolutions of singularities in characteristic zero, which he plans torefine. In the eighth project, Wlodarczyk will further develop histheory of stratified toroidal varieties, which extends the theory oftoroidal embeddings.Algebraic varieties are basic objects in mathematics; they are sets ofsolutions of systems of algebraic equations. They have foundapplications in areas as diverse as mathematical physics andcryptography. The goal of this project is to further the understandingof these objects. A standard technique involves expressing the objectsby their homological invariants, which are usually more accessible andoften computable. Some of the subproject involve this approach. Theremaining subproject involves finding good models for algebraicvarieties.

该项目将被分成代数几何的各个子项目。在第一个子项目中，阿拉普拉打算研究消失定理，这是一组非常粗略的控制同调不变量的技术。在第二个子项目中，AraPurainn倾向于研究与Hodge猜想有关的一些问题，该猜想预测某些被称为Hodge类的同调实体可以以几何方式实现。在第五个子项目中，Matsuki计划研究最小模型程序的同调方面，并研究三维空间的自同构。有几个子项目涉及到为各种不同和它们之间的地图寻找好的模型或分辨率。对于地图，这可以更准确地表述为环面问题，松木将在第四个子项目中研究这一问题。为各种模型找到好的模型相当于解决了奇点。松木和Wlodarczyk将在第七个子项目中研究这一问题的各个方面。特别是，Wlodarczyk已经找到了一种简化的算法来解决特征零点中的奇点，他计划对其进行改进。在第八个项目中，Wlodarczyk将进一步发展他的分层环状簇理论，它扩展了环状嵌入理论。代数簇是数学中的基本对象；它们是代数方程组的解的集合。它们在数学物理和密码学等领域都有广泛的应用。这个项目的目标是加深对这些物体的理解。一种标准的技术涉及到用它们的同调不变量来表示对象，这些不变量通常更容易被访问并且通常是可计算的。一些子项目涉及这一方法。维持子项目包括为代数族找到好的模型。