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.

该项目将分为代数测定法中的各种次要标准。在第一个子项目中，Arapura打算研究消失的理论，这些理论大致是用于控制同源的技术的集合。在第二次副标题中，Arapuraintend研究了与Hodge猜想有关的一些问题，这表明某些称为Hodge类别的同源实体可以以几何方式进行bereal。在第五个子项目中，Matsuki计划了最小模型计划的同源方面，而Alsoto研究了三维空间的自动形态。 SeveralSubproject涉及找到良好的模型或分辨率，以获取它们之间的品种和地图。对于地图，这可以更精确地构成环形化问题，而Matsuki将在第四个副本中研究。寻找各种奇异性的良好模型。第七个子项目中的Matsuki和Wlodarczyk将批准问题的各个方面。 wlodarczyk内部发现了一种简化的特征零奇异性算法前景，他计划了托雷芬。在第八个项目中，wlodarczyk将进一步开发出分层环形品种的理论，该品种扩展了肉芽体嵌入理论。代理品种是数学中的基本对象。它们是代数方程式系统的集合。 他们在像数学物理和递归一样多样化的领域发现了应用。该项目的目的是进一步了解这些对象。标准技术涉及通过其同源不变的对象表达，通常更易于访问和计算。某些子项目涉及这种方法。 介绍的子项目涉及为代数的良好模型寻找良好的模型。