Invariants of Singularities in Zero and Positive Characteristic

零特征和正特征中奇点的不变量

• 批准号: 1068190
• 负责人: Mircea Mustata
• 金额: $ 29.06万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068190&HistoricalAwards=false
• 关键词: Invariants; Singularities; Zero; Positive; Characteristic

The project concerns two questions about singularities. The first problem has two different incarnations: a first one in the algebro-geometric setting of graded sequences of ideals, and a second one, in the analytic setting of plurisubharmonic functions (in which context it was conjectured by Demailly and Koll\'{a}r, and it is known as the Openness Conjecture). The common point is that both versions reduce to understanding asymptotic versions of familiar invariants of singularities, such as the log canonical threshold, associated now to certain sequences of ideals. The first part of the project consists in the study of these asymptotic invariants, and of their connections to valuation theory. The second part will be devoted to a study of valuations from a point of view relevant to this problem. The second problem concerns connections between certain invariants of singularities in birational geometry (such as the log canonical threshold and multiplier ideals) and invariants introduced in commutative algebra, coming from tight closure theory (such as the F-pure threshold and the test ideals). There have been formulated precise conjectures regarding this correspondence via reduction to prime characteristic. This project concerns translating such conjectures into some more established questions regarding the Frobenius action on the cohomology of reductions of smooth projective varieties to positive characteristic, and then in trying to attack some special cases of these questions.Singularities appear naturally in the study of algebraic varieties, and a good understanding of singularities is important, for example, in the classification of higher-dimensional algebraic varieties. This project addresses two important open problems related to singularities. By reducing them to questions in different settings, the PI hopes to bring into action tools from other areas, such as valuation theory and arithmetic geometry. The reduction itself would be of interest, by highlighting connections between these different settings.

该项目涉及有关奇点的两个问题。第一个问题具有两个不同的化身：在plurisubharmonic函数的分析设置中，在代数的几何设置中，第一个问题是一个问题（在demailly和koll \'}的背景下，第二个是一个化身。普遍的观点是，这两个版本都简化为理解熟悉的奇异性不变性的渐近版本，例如对数规范阈值，现在与某些理想序列相关联。该项目的第一部分在于对这些渐近不变的研究及其与评估理论的联系。 第二部分将从与此问题相关的观点开始研究估值。第二个问题涉及在birational几何形状（例如原木规范阈值和乘数理想）中的某些不变性之间的联系与交换代数中引入的不变式之间的连接，来自紧密的封闭理论（例如F-Pure阈值和测试理想）。通过还原到主要特征，已经有关于这种对应关系的精确猜想。 This project concerns translating such conjectures into some more established questions regarding the Frobenius action on the cohomology of reductions of smooth projective varieties to positive characteristic, and then in trying to attack some special cases of these questions.Singularities appear naturally in the study of algebraic varieties, and a good understanding of singularities is important, for example, in the classification of higher-dimensional algebraic varieties.该项目解决了与奇点有关的两个重要开放问题。通过将它们减少到不同环境中的问题，PI希望从其他领域（例如估值理论和算术几何形状）带入行动工具。通过强调这些不同设置之间的联系，减少本身将引起人们的关注。