Invariants of Singularities in Zero and Positive Characteristic

该项目涉及两个关于奇点的问题。第一个问题有两个不同的化身：第一个问题在分次理想序列的代数几何环境下，第二个问题在多重亚调和函数的解析环境下(在这种情况下，它是由Demailly和Koll‘a}r猜想的，称为开性猜想)。共同点是，这两个版本都归结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的对数正则阈值。该项目的第一部分在于研究这些渐近不变量，以及它们与估值理论的联系。第二部分将致力于从与这个问题相关的角度来研究估值。第二个问题是关于二次几何中奇点的某些不变量(如对数正则门限和乘子理想)与紧闭包理论(如F-纯门限和检验理想)在交换代数中引入的不变量之间的关系。通过约化为素数特征，已有关于这种对应关系的精确猜想。这个项目涉及将这样的猜想转化为关于光滑射影簇的上同调到正特征的Frobenius作用的一些更确定的问题，然后试图攻击这些问题的一些特殊情况。奇点在代数簇的研究中自然地出现，而对奇点的良好理解是重要的，例如，在高维代数簇的分类中。这个项目解决了与奇点有关的两个重要的公开问题。通过将它们简化为不同背景下的问题，PI希望将其他领域的工具带入行动，如估值理论和算术几何。通过强调这些不同设置之间的联系，这种减少本身将是有意义的。

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.