Local and global problems on singularities for higher dimensional algebraic varieties

高维代数簇奇点的局部和全局问题

• 批准号: 0700360
• 负责人: Nero Budur
• 金额: $ 9.94万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700360&HistoricalAwards=false
• 关键词: Local; global; problems; singularities; higher

The proposed research is in the general area of algebraic geometry. The proposed projects investigate the complexity of singularities of algebraic varieties from both the local and the global point of view. Concerning the local theory of singularities, the PI proposes to investigate relations among different approaches to singularities: resolutions of singularities, Hodge theory and D-modules, and jet schemes. The first project considers restrictions on multiplier ideals induced by Hodge theory. Employing the use of D-modules and combinatorial techniques, the PI plans to study the subtle nature of an invariant of singularities called the Bernstein-Sato polynomial. In a third project, the PI looks for the connection between D-modules and jet schemes. The approach is to construct a notion of motivic integration with values in a derived category of D-modules. Concerning the global theory, the PI together with L. Ein plans to find criteria for geometric stability in terms of multiplier ideals. Such criteria would be important for the study of extremal metrics in Kahler geometry and would involve global cohomological invariants of singularities in terms of multiplier ideals. In the last project proposed, the PI studies a natural setting for these global cohomological invariants in terms of local systems.Algebraic geometry is the study of solutions of algebraic equations.It is one of the modern successors of ancient Greek geometry in the sense that geometrical shapes have been replaced by the equations defining them. The richest and most complicated structure of these geometrical shapes appears at certain places called singularities. A local study of singularities is roughly like placing them under a microscope and zoomming on them. A global study is to understand the restrictions imposed by the singularities on the behavior of the geometrical shape with respect to other geometrical shapes.

拟议的研究是在一般领域的代数几何。拟议的项目调查的复杂性，奇异的代数簇从本地和全球的角度来看。关于奇点的局部理论，PI建议研究奇点的不同方法之间的关系：奇点的解决方案，霍奇理论和D-模块，以及喷气计划。第一个项目考虑霍奇理论对乘数理想的限制。利用D-模和组合技术，PI计划研究称为Bernstein-Sato多项式的奇异性不变量的微妙性质。在第三个项目中，PI寻找D模块和喷气式飞机方案之间的联系。该方法是构造一个概念的motivic整合的价值观在一个派生范畴的D-模。关于整体理论，PI与L.艾因计划寻找乘子理想的几何稳定性准则。这样的准则对于卡勒几何中的极值度量的研究非常重要，并且将涉及奇异点在乘子理想方面的全局上同调不变量。在最后一个项目中，PI研究了这些全局上同调不变量在局部系统中的自然设置。代数几何学是研究代数方程的解的学科。它是古希腊几何学的现代继承者之一，因为几何形状已经被定义它们的方程所取代。这些几何形状的最丰富和最复杂的结构出现在某些称为奇点的地方。对奇点的局部研究大致上就像把它们放在显微镜下放大一样。全局研究是为了理解奇点对几何形状相对于其他几何形状的行为所施加的限制。