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.

该项目涉及两个有关奇点的问题。第一个问题有两种不同的体现：第一个是在理想分级序列的代数几何设置中，第二个是在多次调和函数的解析设置中（在这种情况下，它是由德梅利和科尔推测的\'{a }r，它被称为开放性猜想）。共同点是，这两个版本都简化为理解熟悉的奇点不变量的渐近版本，例如现在与某些理想序列相关联的对数规范阈值。该项目的第一部分包括研究这些渐近不变量及其与估值理论的联系。 第二部分将从与该问题相关的角度进行估值研究。第二个问题涉及双有理几何中奇点的某些不变量（例如对数正则阈值和乘数理想）与来自紧闭包理论的交换代数中引入的不变量（例如 F-纯阈值和测试理想）之间的联系。通过简化为素数特征，已经对这种对应关系提出了精确的猜想。 该项目涉及将此类猜想转化为一些关于平滑射影簇还原到正特征的上同调的弗罗贝尼乌斯作用的更确定的问题，然后尝试解决这些问题的一些特殊情况。奇点在代数簇的研究中自然出现，并且对奇点的良好理解很重要，例如，在高维代数簇的分类中。该项目解决了与奇点相关的两个重要的开放性问题。通过将它们简化为不同环境中的问题，PI 希望将其他领域的工具（例如估值理论和算术几何）付诸实践。通过强调这些不同环境之间的联系，减少本身就很有趣。