Abstract:
In this talk, I will first introduce notions (rational, log terminal, log canonical singularities, and multiplier ideals) arising from birational geometry in characteristic 0. I will then turn to F-singularities (F-rationality, strong F- regularity, and F-purity) and test ideals, which are defined using the Frobenius morphism in positive characteristics and are important concepts in tight closure theory.
The fascinating connections between these two areas have been discovered and explored for decades. Many theorems in one area were motivated by the ideas (or even proved by the techniques) from the other area.
The last part of this talk is to illustrate the powerfulness of these connections by mentioning the following three works of mine.
1. Determinantal varieties are log terminal in the sense of de Fernex and Hacon.
2. New proof of normality and rationality of singularities of Schubert varieties.
3. (jointed with K. Schwede and W. Zhang) The correspondence between Cartier modules and certain generalization of ideals defining unions of log canonical centers on toric varieties.