F-singularities and singularities in birational geometry in characteristic zero

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400039
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: Takagi Shunsuke 東京大学, 大学院数理科学研究科, 准教授 (40380670)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: F特異点 / 可換環論 / 特異点論 / 局所コホモロジー

Outline of Final Research Achievements

``F-singularities" are a generic term used to refer to singularities defined in terms of Frobenius maps, and there are four basic classes of F-singularities, F-regular, F-pure, F-rational and F-injective singularities. F-singularities are expected to correspond to the singularities in birational geometry in characteristic zero, and many researchers have studied this correspondence. In this research project, we obtained the following two results related to this correspondence: (1) When X is a numerically Q-Gorenstein variety over an algebraically closed field of characteristic zero, we proved that the multiplier ideals on X (in the sense of de Fernex-Hacon) coincide, after reduction to characteristic p>>0, with the test ideals. (2) We introduced a new class of F-singularities, F-nilpotent singularities, and gave a Hodge theoretic interpretation of (3-dimensional) F-nilpotent normal isolated singularities.

Free Research Field

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