Liftings of ideals in positive characteristic to those in characteristic zero: Higher dimension

We study a pair consisting of a smooth variety of arbitrary dimension over a field of positive characteristic and a multi-ideal with a real exponent. We prove that the set of log discrepancies for a fixed exponent is discrete. Additionally, we show that the set of log canonical thresholds (lcts) of multi-ideals on a smooth variety in positive characteristic is contained within the set of lcts of multi-ideals on a smooth variety over the complex number field. As a result, we find that the accumulation points of log canonical thresholds are rational if all the exponents are rational. We also obtain ACC for the set of lcts of multi-ideals on a smooth varieties in positive characteristic. These findings generalize the author's results presented in arXiv in 2024 to the higher-dimensional case.