Invariants of Singularities in Zero and Positive Characteristic

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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.